THIS IS THE (CO)END, MY ONLY (CO)FRIEND
FOSCO LOREGIAN†
Abstract. The present note is a recollection of the most striking anduseful applications of co/end calculus. We put a considerable effort
in making arguments and constructions rather explicit: after having
given a series of preliminary definitions, we characterize co/ends as
particular co/limits; then we derive a number of results directly from thischaracterization. The last sections discuss the most interesting examples
where co/end calculus serves as a powerful abstract way to do explicit
computations in diverse fields like Algebra, Algebraic Topology andCategory Theory as well as some generalizations to higher dimensional
category theory.
The appendices serve to sketch a number of results in theories heavily
relying on co/end calculus; the reader who dares to arrive at this point,
being completely introduced to the mysteries of co/end fu (端楔術,duānxiē shù; literally “the art [of handling] terminal wedges”), canregard basically every statement as a guided exercise.

Contents
Introduction.
1. Dinaturality, extranaturality, co/wedges.
2. Yoneda reduction, Kan extensions
3. The nerve and realization paradigm.
4. Weighted limits.
5. The theory of relators.
6. Operads using coends.
7. Higher dimensional coend calculus.
Appendix A. Promonoidal categories
Appendix B. Fourier transforms via coends.
References

2
6
20
24
35
46
55
63
91
93
94

Key words and phrases. end, coend, dinatural transformation, operad, profunctor,
relator, Kan extension, weighted limit, nerve and realization, promonoidal category, Yoneda
structure.
12

F. LOREGIAN

Introduction.
The purpose of the present survey is to familiarize its readers with what
in category theory is called co/end calculus, gathering a series of examples
of its application; the author would like to stress clearly, from the very
beginning, that the material presented here makes no claim of originality:
we put a special care in acknowledging carefully, where possible, each of
the countless authors whose work was a source of inspiration in compilingthis note. Among these, every erroneous or missing attribution must be
ascribed to the mere ignorance of the author, which invites everyone hedidn’t acknowledge to contact him.
The introductory material appearing in section 1 is the most classicaland comes almost verbatim from the classical reference [ML98]; the nerverealization formalism is a patchwork of various results, scattered in the(algebraic) topology literature; these results are often presented as a deus
ex machina, mysterious machineries leaving the reader without any clue
about why they work so well. The reformulation of operads using coends
comes verbatim from [Kel05]; the chapter discussing the theory of relators (orprofunctors, or distributors; in French it is common to call them distributeurs,
following an idea of J. Bénabou; we deviate from this unfortunate use) comesfrom the work of J. Bénabou, and I immensely profited from some notes taken
by T. Streicher, [Béner]; during May 2016 I had the pleasant opportunity
to meet Thomas, and I fell in love with his mathematical style, near to thecraftsmanship of certain watchmakers, but also vital and passionate. I hope
this text is better, in view of those pleasant days of friendly mathematics.
Chapter 4 on weighted co/limits is taken almost verbatim from [Rie14,II.7]: only a couple of implicit conceptual dependencies on co/end calculus
have been made explicit, and some examples left as exercises there are
discussed in full detail; the fact that we follow so faithfully the exposition of[Rie14] must be interpreted not as an act of plagiarism, but instead ad an
implicit invitation to get acquainted with such a wonderful book. I countEmily as one of the most fierce supporters of Category Theory among the“new generation”, and I’m proud to be considered a good mathematician byher.
A subsequent section discusses co/end calculus in higher category theory,
introducing lax co/ends in 2-categories [Boz77, Boz75, Boz80], homotopy
ends [DF78, Isa09] and (∞, 1)-co/ends in various models: first of all JoyalLurie’s ∞-categories following [GHN15], and then Bergner’s simpliciallyenriched categories mainly following [CP97] and the general theory forenriched co/ends in [Dub70, Gra].
The relation between co/end calculus and homotopy co/limits is discussed,
and a(n already) classical result [Gam10] is presented to unify the two

ENDS

3

constructions classically given for the homotopy co/limit of a´diagram; thecompatibility of the co/end operation as a Quillen functor : Cat(Cop ×C, D) → D is discussed when´ D is a sufficiently nice model category (ourmain reference to prove that is a Quillen functor is [Lur09, A.2.9.28]).
The last chapter about promonoidal categories comes from [Day74, Str12];
the subsequent appendices propose the reader to familiarize with the theoryof promonoidal categories, and to show the initial results of [Day11, §1-3],
a delightful and deep paper whose importance is far more than a source of
unusual exercises.
Because of all these remarks, the reader has to keep in mind that thevalue of this work –if there is any– lies, rather than in the originality of the
discussion (which is affected by a series of unforgivable sins of omission), in
its strong will to be simple and clear remaining an exhaustive account, collecting as much material as possible without leaving obscure or unreferenced
passages.
Our aim is to serve the inexperienced reader (either the beginner in the
study of category theory, or the experienced student who exhausted the“primary” topics of her education) as a guide to familiarize with this extraordinarily valuable tool-set producing a large number of abstract-nonsense
proofs, most of which are “formally formal” strings of natural isomorphisms.
It is our firm opinion that this document could serve as a pedagogical tool
even for some experienced mathematicians, maybe working nearby category
theory, that are never been exposed to this beautiful machinery. It is ourhope that they could find their way to exploit such a powerful language.
After the first examples, the keen reader will certainly prefer to re-write
most of the proofs in the silence of her room, and we warmly invite her todo so; imitation of the basic techniques is for sure an unavoidable step ingetting acquainted with the machinery of co/end calculus, and more generallywith any machinery whatsoever. This is especially true for mathematicallanguage, in the same sense we all learned integration rules by imitation andtraining reading precalculus books.
Maybe it’s not a coincidence that the one you are about to see is anotherintegral calculus to be learned by means of examples and exercises. This
analogy could be pushed further; we refrain to do it, lest you think we arearrogantly claiming to be able to reduce the subtle art of integration tocategory theory. Nevertheless, we can’t help but mention several insightfulformal and informal analogies between mathematical analysis and co/end
calculus: these are scattered throughout all the discussion, and we denotethem with the special symbol .1 Whenever it appears, we advise the reader
1I learned this funny notation during my freshman year, when I was handed [DM96] forthe first time; the “small-eyes” notation accompanied me throughout all my mathematicallife until today. Various (facial) expressions advise different ways the reader is supposed to4

F. LOREGIAN

feeling uncomfortable with a certain dose of hand-waving and categorical
juggling to raise her eyes and skip the paragraph.
It has been said that “Universal history is, perhaps, the history of a few
metaphors” [Bor64]: differential and integral calculus is undoubtedly oneof such recurring themes. In some sense, the fortune of co/end calculus isbased on the analogy which represents these universal objects by means ofan integral symbol [Yon60, DK69, Str12]; this analogy is motivated by the
Fubini theorem on the interchange of “iterated integrals”: unfortunately,
despite its immense expressive and unifying power the language of coends
seems to be woefully underestimated. No elementary book in category
theory (apart a single chapter in the aforementioned [ML98]) seems tocontain something more than a bare introduction of the basic elements of
the language.
What’s missing, in the humble opinion of the author, is an exhaustive
and unitary source of examples, exercises and computations, showing theirreaders how co/end calculus can literally disintegrate involved computations
and reduce them to a bunch of canonical isomorphisms.
Trying to fill this gap in the literature has been the main motivation for
the text you’re about to read. It’s up to you to decide it it is a clumsy
attempt, or a partial success.
A final note, January 3, 2017. It is impossible, at this point, to keep
track of all the influences I received since when this project started. The
present document doubled its length since its first version appeared on thearXiv. Probably someday a “version 3” will appear. Thanks everybody,especially to the people I acknowledge below.
Acknowledgements. In some sense, I am not the only, and for sure
I’m the youngest and less experienced, author of this note. I would liketo thank T. Trimble, E. Rivas and A. Mazel-Gee for having read carefully
the preliminary version of this document, suggesting improvements andcorrections, having spotted a disgraceful number of errors, misprints andincoherent choices of notation. Their attentive proofreading has certainly
increased the value –again, if there is any– of the document you’re about toread.
This humongous amount of errors did not prevent the first version ofcoend-cofriend to circulate, and being read, far more than we expected. And
this is true to the point that at the moment of writing this is the paperwith the highest number of (formal or moral) citations. I warmly thank the, abstract material;,
behave when she meets them. There are four such expressions:standard exercises;, material that you are supposed to meditate a lot;, shattering
exercises.

ENDS

5

people who judged this clumsy list of examples a document worth to beread, improved and even studied.
I want to thank the people who supported directly and indirectly thegenesis of this paper: a conversation with A. Joyal in a café in Paris, where
I wrote the statement of Example 4.22 on a napkin to motivate the ubiquity
and supremacy of coend-fu, happened in June 2013 and convinced me tostart the project; the f&h colleagues and friends A. Gagna, E. Lanari, G.
Mossa, F. Genovese, M. Vergura, I. Di Liberti, S. Ariotta, G. Ronchi enduredsometimes unpleasant conversations on “why every mathematician should
know co/end calculus”; D. Fiorenza, the advisor my whole life of devotion
will not be enough to refund for his constant, invaluable friendship andsupport, spurred me to turn a series of chaotic sheets of paper into thepresent note; N. Gambino offered me the opportunity to discuss the content
of this note in front of his students in Leeds; in just a few days I realized
years and years of meditation were still insufficient to teach this subject.
L. accompanied me there, making me sure that she’s the best mate for amuch longer trip. S., P., G., C.opened me their doors when I was frail andbroken-hearted. Grazie.
Foundations, notation and conventions. The main foundational convention we adopt throughout the paper is the assumption ([GV72]) thatevery set lies in a suitable Grothendieck universe. We implicitly fix suchan universe , whose elements are termed sets; categories are always considered to be small with respect to some universe: in particular we choose
to adopt, whenever necessary, the so-called two-universe convention, where
we postulate the existence of a universe + ⊃ in which all the non- small categories live. This rather common choice has nevertheless subtle
consequences: as it is recorded in [Wat75, Low13] the existence and good
behaviour of some co/limits and Kan extensions critically depends from theparticular choice of a universe.
At least in some situations, it is still possible to keep this problem undercontrol, appealing some “boundedness” conditions keeping track of the
cardinality of the involved constructions. There are few places where thiscaveat could become a real problem (like for example our 3.9).
Several kinds of categorical structures (categories, and often also 2-categories and bicategories, as well as instances of higher categories) will bedenoted as boldface letters C, D, . . . ; the context always clarifies which
structure is considered each time. Functors between categories are denoted
as capital Latin letters like F, G, H, K and suchlike (there can be little
deviations to this rule, like for example in §5); the category of functors
C → D between two categories is denoted as Fun(C, D), DC , [C, D] and suchb is a shorthand for the category
like; following a long-standing tradition, C
Ω

Ω

Ω

Ω

6

F. LOREGIAN[Cop , Sets] of presheaves on C; the canonical hom-bifunctor of a category C
sending (c, c0 ) to the set of all arrows hom(c, c0 ) ⊆ hom(C) is almost always
denoted as C( , ) : Cop × C → Sets, and the symbols ,are used asplaceholders for the “generic argument” of a functor or bifunctor; morphisms
in the category Fun(C, D) (i.e. natural transformations between functors)
are often written in Greek, or Latin lowercase alphabet, and collected in theset Nat(F, G) = DC (F, G). The simplex category ∆ is the topologist’s delta(opposed to the algebraist’s delta ∆ + which has an additional initial object[−1] , ∅), having objects nonempty finite ordinals [n] , {0 < 1 · · · < n}; wedenote ∆[n] the representable presheaf on [n] ∈ ∆ , i.e. the image of [n] underb of simplicial sets. Morethe Yoneda embedding of ∆ in the category sSet = ∆generally, we indicate the Yoneda embedding of a category C into its presheaf
category with よC –or simply よ–, i.e. with the hiragana symbol for “yo”;
this choice comes from [LB15], and I share a similar aesthetics for peculiar
notation. Whenever there is an adjunction F a G between functors, thearrow F a → b in the codomain of F and the corresponding arrow a → Gb in
its domain are called mates or adjuncts; so, the notation “the mate/adjunctof f : F a → b” means “the unique arrow g : a → Gb determined by f ”.
Los idealistas arguyen que las salas hexagonales son
una forma necesaria del espacio absoluto o, por lomenos, de nuestra intuición del espacio.

J.L. Borges, La biblioteca de Babel.

1. Dinaturality, extranaturality, co/wedges.
Let’s start with a simple example. Let Sets be the category of sets andfunctions, considered with its natural cartesian closed structure: this means
we have a bijection of sets
Sets(A × B, C) ∼(1)
= Sets(A, C B )where C B is the set of all functions B → C. The adjunction
has a counit, which is a natural transformation
X,(B) : X B × B → X

× B a ( )B(2)

where the codomain can be considered “mutely depending on the variable B”.
The collection of functions {X : X B × B → X} is natural in the classicalsense in the variable X; as for the variable B, the most we can say is thecommutativity above: it doesn’t remind naturality so much. See also Exercise
2 at the end of [ML98, IV.7] and our Exercise 1.E1: the naturality in B of
the adjunction Sets(A × B, C) ∼= Sets(A, C B ) implies a “more complicated”
dependence on the variable B.

ENDS

7

Fortunately a suitable generalization of naturality (a “super-naturality”
condition), encoding the result of Exercise 1.E1, is available to describe thisand other similar phenomena in the same common framework. A notion ofsuper-naturality adapted to describe co/ends as suitable universal objects
comes in two flavours: one of them is dinaturality, which we now introduce.
Definition 1.1 [Dinatural Transformation]: Given two functors P, Q : Cop ×..
→ Q, consists
C → D a dinatural transformation, depicted as an arrow α : P −of a family of arrows {αc : P (c, c) → Q(c, c)}c∈C such that for any f : c → c0
the following hexagonal diagram commutes
P (f,c)P (c0 , c)
P (c0 ,f )P (c0 , c0 )αc0

/ P (c, c)/ Q(c0 , c0 )αc

/ Q(c, c)Q(c,f )(3)

/ Q(c, c0 )Q(f,c0 )Definition 1.2 [Wedge for a functor]: Let P : Cop × C → D; a wedge for..
P is a dinatural transformation ∆d −→ P from the constant functor on theobject d ∈ D (often denoted simply by d : Cop × C → D), defined sending(c, c0 ) 7→ d, (f, f 0 ) 7→ idd .
Definition 1.3 [End of a functor]: The end of a functor F : Cop × C → D..
consists of a universal wedge end(F ) −→ F ; the constant end(F ) ∈ D is itself
termed, by abuse, the end of the functor.
Spelled out explicitly, the universality requirement means that for any..
other wedge β : d −→ F the diagram
d

βc
h

#

end(F )
βc0

ωc0

"

ωc



F (c0 , c0 )$/ F (c, c)F (f,1)F (1,f )/ F (c, c0 )

(4)

c


f

c0

commutes for a unique arrow h : d → end(F ), for any arrow f : c → c0 .
Remark 1.4 : Uniqueness requirements imply functoriality: given a naturaltransformation η : F ⇒ F 0 there is an induced arrow end(η) : end(F ) →8

F. LOREGIAN

end(F 0 ) between their ends, as depicted in the diagramend(F 0 )/ F 0 (c0 , c0 )88end(F )/ F (c0 , c0 )


(5)


F 0 (c, c)/ F 0 (c, c0 )

8

8





F (c, c)/ F (c, c0 )This implies that taking the end of a functor is a (covariant) functorop
DC ×C → D. The case of a coend is dually analogous: filling the details isan easy dualization exercise.
A slightly less general, but better behaved notion of super-naturality(we say “better behaved” since it admits a graphical calculus translating
commutativity theorems into controlling that certain string diagrams can be
deformed one into the other), allowing again to define co/wedges, is available:
this is called extra-naturality and it was introduced in [EK66].
Definition 1.5 [Extranatural transformation]: Let P, Q be functors
P : A × Bop × B → D

Q : A × Cop × C → D.
An extranatural transformation α : P −→ Q consist of a collection of arrows..

αabc : P (a, b, b) −→ Q(a, c, c)(6)indexed by triples of object in A × B × C such that the following hexagonal
diagram commutes for every f : a → a0 , g : b → b0 , h : c → c0 , all taken in
their suitable domains:
P (a, b0 , b)
P (a,g,b)

P (f,b0 ,g)

P (a, b, b)

αabc0/ P (a0 , b0 , b0 )/ Q(a, c0 , c0 )αa0 b0 c/ Q(a0 , c, c)Q(f,h,c0 )Q(a0 ,c,h)(7)/ Q(a0 , c, c0 );This commutative hexagon can be equivalently described as the juxtaposition of three distinguished commutative squares, depicted in [EK66]: thethree can be obtained letting respectively f and h, f and g, or g and h be

ENDS

9

identities in the former diagram, which collapses to
P (a, b, b)
αabc

Q(a, c, c)

P (a,b0 ,g)

P (f,b,b)/ P (a0 , b, b)

αa0 bcP (a, b0 , b)
P (a,g,b)/ Q(a0 , c, c)

Q(f,c,c)/ P (a, b0 , b0 )P (a, b, b)



αabc

αab0 c/ Q(a, c, c)

αabc

P (a, b, b)
αabc0





Q(a, c0 , c0 )Remark 1.6 : We can again define co/wedges in this setting: if B = C and in
F (a, b, b) → G(a, c, c) the functor F is constant in d ∈ C, G(a, c, c) = Ḡ(c, c)..
is mute in a, we get a wedge condition for d −→ G; dually we obtain a cowedge
condition for F (b, b) → G(a, b, b) ≡ d0 for all a, b, c.

Both notions give rise to the same notion of co/end as a universal co/wedge
for a bifunctor F : Cop × C → D. The main reason we should prefer extranaturality, as was pointed out to the author in a (semi)private conversation
with T. Trimble, is that
For the purposes of describing end/coend calculus, I wouldn’t emphasize dinatural transformations so much as I would extranaturaltransformations. Most dinatural transformations that arise in
the wild can be analyzed in terms of extranatural (extraordinary
natural, in the old lingo) transformations. [. . . ][Co/wedges can be regarded as particular examples of twoslightly different, but related (see Prop 1.8) constructions:] first,
they are special examples of dinatural transformations. Second,
they are special cases of an extranatural (extraordinary natural)transformation, which generally is a family of maps F (a, a, b) →G(b, c, c) which combines naturality in the argument b with acowedge condition on a and a wedge condition on c.

We now briefly describe the promised graphical calculus for extranaturaltransformations: it depicts the components αabc , and arrows f : a → a0 ,g : b → b0 , h : c → c0 , as planar diagrams likeG(a, c, c)

a

b

c

f

g

h

a0

b0

c0

Q(a,c,h)/ Q(a, c, c0 )Q(a,h,c0 )(8)F (a b, b)/ Q(a, c, c)

where wires are labeled by objects and must be thought oriented from top
to bottom. The commutative squares of (8) become, in this representation,
the following three string diagrams, whose equivalence is graphically obvious(the labels f, g, h can “slide” along the wire they live in):

10

F. LOREGIAN

f

=

f

g

=

g

=
h

h

Remark 1.7 : The notion of extranatural transformation can be specialized
to encompass various other constructions: simple old naturality arises when
F, G are both constant in their co/wedge components, like inwhereas wedge and cowedge conditions arise when either F, G are constant:

All the others mixed situations (a wedge-cowedge condition, naturality anda wedge, etc. which lack a specified name) admit a graphical representationof the same sort, and follow similar graphical rules of juxtaposition.
Proposition 1.8 : Extranatural are particular kinds of dinatural transformations.
Proof (due to T. Trimble). Suppose you have functors F : Cop × C × C → D
and G : C × C × Cop → D. Now put A = C × Cop × Cop , and form two new
functors F 0 , G0 : Aop × A → D by taking the composites
F 0 = (Cop × C × C) × (C × Cop × Cop ) −−→ Cop × C × C −
→D
proj

F

(x0 , y 0 , z 0 ; x, y, z) 7−→ (x0 , x, y 0 ) 7→ F (x0 , x, y 0 )
F

proj0

G0 = (Cop × C × C) × (C × Cop × Cop ) −−−→ C × C × Cop −→DG(x0 , y 0 , z 0 ; x, y, z) 7→ (y 0 , z 0 , z) 7→ G(y 0 , z 0 , z)G

Now let’s put a0 = (x0 , y 0 , z 0 ) and a = (x, y, z), considered as objects in A.
An arrow ϕ : a0 → a in A thus amounts to a triple of arrows f : x0 → x,
g : y → y 0 , h : z → z 0 all in C.
Following the instructions above, we have F 0 (a0 , a) = F (x0 , x, y 0 ) andG(a0 , a) = G(y 0 , z 0 , z).

ENDS

11

Now if we write down a dinaturality hexagon for α : F 0 → G0 , we get a
diagram of shape..

F 0 (a, a0 )F (ϕ,1)F 0 (1,ϕ)

F 0 (a0 , a0 )αa

/ F 0 (a, a)/ G0 (a0 , a0 )αa0

/ G0 (a, a)
G0 (ϕ,1)(9)/ G(a0 , a)G0 (1,ϕ)which translates to a hexagon of shape
F (x, x0 , y)F (f,1,g)

F (1,f,1)



F (x0 , x0 , y 0 )/ F (x, x, y)/ G(y, z, z)/ G(y 0 , z 0 , z 0 )

G(g,h,1)(10)/ G(y 0 , z 0 , z)G(1,h,1)

where the unlabeled arrows refer to the extranatural transformation.



1.1. The integral notation for co/ends. A suggestive and useful notation alternative to the anonymous one “co/end(F )” is due to N. Yoneda,
which in [Yon60] introduces most of the notions we are dealing with, specialized to Ab-enriched functors Cop × C → Ab: the integral notation
denotes´
opthe end of a functor F ∈ DC ×C as a “subscripted-integral”(c,c), andFc´c
F (c, c).
the coend coend(F ) as the “superscripted-integral”
From now on we will systematically adopt this notation to denote the universal co/wedge co/end(F ) or, following a well-established abuse of notation,
the object itself; we also accept slightly more pedantic variants of this, asˆ /c∈Cˆ /c:CF (c, c),F (c, c).
(11)
c∈C

c:C

Remark 1.9 : One should be aware that Yoneda’s notation in´ [Yon60] is“reversed” as he calls integration our coends, which he denotes as c∈C F (c, c),´∗
and cointegrations our ends, which he denotes c∈C F (c, c).
Remark 1.10 : Properties of co/ends acquire a particularly suggestive flavour
when written in this notation:
1) Functoriality (the B-rule for integrals or the freshman’s dream): theunique arrow end(η) induced by a natural transformation
η´: F ⇒ G´´opbetween F, G ∈ DC ×C can be written as ´η : F →´ G,´ anduniqueness of this induced arrow entails that (η ◦ σ) = η ◦ σ.
2) The “Fubini theorem” for ends, which first appeared as equation(4.0.1) of [Yon60]: given
a functor F : Cop × C × Eop × E → D, we´
can form the end c F (c, c, , ) obtaining a functor Eop × E →12

F. LOREGIAN´ ´
Dthe ends´ ´whose end is e c F (c, c,´ e, e) ∈ D; we can also form op
F
(c,
c,
e,
e)
∈
D
and
F
(c,
c,
e,
e)
identifying
C
×C×
c e
(c,e)
op
op
E × E with (C × E) × (C × E). Fubini’s theorem for ends statesthat there is a canonical isomorphism between the three:

ˆ

(c,e)

F (c, c, e, e) ∼
=

ˆ ˆ
e

c

F (c, c, e, e) ∼
=

ˆ ˆ
c

F (c, c, e, e)

(12)

e

Remark 1.11 : In some sense, the Fubini rule for coends seems a rather weakanalogy between integrals and coends; indeed there is no doubt that thefollowing passage ([ML98, IX.5])´[. . . ] the “variable of integration” c [in c F ] appears twice
under the integral sign (once contravariant, once covariant)and is “bound” by the integral sign, in that the result nolonger depends on c and so is unchanged if c is replaced byany other letter standing for an object of the category C
motivates the integral notation, and nevertheless, this analogy seems quite
elusive to justify in a precise way. In the eye of the author, it seems
worthwhile to remember that in view´of the characterization for co/ends inopterms of co/equalizers given in 1.15, c : DC ×C → D can be thought as anaveraging operation on a functor, giving the “fixed points” of the “action”
induced by F (ϕ, c0 ), F (c, ϕ) as ϕ : c → c0 runs over hom(C).
Although the author prefers to abstain from any further investigation,
having no chance to give a valid (or rather, formal) explanation of theintegral notation, it is nevertheless impossible to underestimate the powerof this convenient shorthand.
1.2. Co/ends as co/limits. A general tenet of elementary category theoryis that you can always characterize a universal construction as an element of
the triad
limit - adjoint - representation of functors.
The formalism of co/ends makes no exception: the scope of the following
subsection is to characterize, whenever it exists, the co/end of a functor
F : Cop × C → D as a co/limit over a suitable diagram, and finally as theco/equalizer of a single pair of arrows.

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13

First of all notice that given F : Cop × C → D and a wedge τ : d −→ F , wecan build the following commutative diagram
..

τc

d

8

F (c, c)F (c,f )τc 0/ F (c0 , c0 )

τc00

τgf



F (c00 , c00 )F (g,c00 )/ F (c, c0 )7
F (f,c0 )F (c,g)
0 F (c, c00 )
7

(13)


F (f,c00 )/ F (c0 , c00 )where c −→ c0 →− c00 are two arbitrarily chosen, but fixed, arrows in C. From
this commutativity we deduce the following relations:fg

τgf = F (gf, c00 ) ◦ τc00 = F (c, gf ) ◦ τc

= F (f, c00 ) ◦ F (g, c00 ) ◦ τc00 = F (f, c00 ) ◦ τg

= F (c, g) ◦ F (c, f ) ◦ τc = F (c, g) ◦ τf .

where τf , τg are the common values F (f, c0 )τc0 = F (c, f )τc and F (c0 , g)τc0 =F (g, c00 )τc00 respectively, τgf is the common value F (c, g)τf = F (f, c00 )τg .
These relations imply that there is a link between co/wedges and co/cones,
encoded in the following definition.2
Definition 1.12 [The twisted arrow category of C]: For every category
C we define tw(C), the category of twisted arrows in C as follows:• Ob(tw(C)) = hom(C);• Given f : c → c0 , g : d → d0 a morphism f → g is given by a pair ofarrows (h : d → c, k : c0 → d0 ), such that the obvious square commutes(asking that the arrow between domains is in reversed order is not a
mistake!).
Endowed with the obvious rules for composition and identity, tw(C) iseasily seen to be a category, and now we can find a functor
Fun(Cop × C, D)/ Fun(tw(C), D)(14)

defined sending F to the functor F : tw(C) → D : f 7→ F (src(f ), trg(f )); it
is extremely easy to check that bifunctoriality for F exactly corresponds tofunctoriality for F , but there is more.
2I am indebted to Giorgio Mossa (Università degli studi di Pisa, e chissà per quanto)for having revealed me this argument when I was still unable to manipulate co/ends.

14

F. LOREGIAN

Remark 1.13 : The family {τf }f ∈hom(C) constructed before is a cone for thefunctor F , and conversely any such cone determines a wedge for F , givenby {τc , τidc }c∈C . Again, a morphism of cones goes to a morphism between
the corresponding wedges, and conversely any morphism between wedges
induces a morphism between the corresponding cones; these operations aremutually inverse and form an equivalence between the category cn(F ) ofcones for F and the category wd(F ) of wedges for F (see Exercise 1.E5).
Equivalences of categories obviously respect initial/terminal objects, andsince co/limits are initial/terminal objects in the category of co/cones, andco/ends are initial/terminal objects in the category of co/wedges, we obtain
that3
ˆ
ˆ
F;
F ∼
= lim
←−
c
tw(C)

c

F ∼
=

lim
−→

F

(15)

tw(Cop )op

Remark 1.14 : There is another (slightly ad hoc and cumbersome, in the
humble opinion of the author) characterization of co/ends as co/limits, given
by [ML98, Prop. IX.5.1], which relies upon the subdivision (associative) plot(see [Trib, Def. 2, 3]) (℘C)§ of C, whose• objects are the set Ob(C) t hom(C), in such a way that there exists
a “marked” object c§ for each c ∈ C, and another marked object f §for each f ∈ hom(C). The reader must have clear in mind that c§and id§c are different objects of C§ ;• arrows are the set of all symbols src(f )§ → f § , or trg(f )§ → f § , as fruns over hom(C);• composition law is the empty function.
The subdivision category C§ is obtained from (℘C)§ formally adding identities
and giving to the resulting category the trivial composition law (composition
is defined only if one of the arrows is the identity). The discussion before[ML98, Prop. IX.5.1] now sketches the proof that every functor F : Cop ×C →D induces a functor F : C§ → D, whose limit (provided it exists) is isomorphic
to the end of F .
Remark 1.15 : Co/limits in a category exist whenever the category has
co/products and equalizers. So we would expect a characterization ofco/ends in terms of these simpler pieces; such a characterization exists,
and it turns out to be extremely useful in explicit computations (see forexample our Remark 5.11 and the argument therein), characterizing co/ends
3Notice that the colimit is taken over the category twop (C), the opposite of tw(Cop ):an object of twop (C) is an arrow f : c0 → c in Cop , and a morphism from f : c → c0 to
g : d → d0 is a commutative square such that vgu = f .

ENDS

15

as co/equalizers of pairs of maps. In fact it is rather easy to see thatc F (c, c) = eq

´

∼




Y

F∗

F (c, c)

F∗

c∈C

/
/

Y

ϕ : c→c0F (c, c0 )

(16)

where the product over ϕ : c → c0 can be expressed as a double product(over the objects c, c0 ∈ C, and over the arrows ϕ between these two fixed
objects), and the arrows F ∗ , F∗ are easily obtained from the arrows whose(ϕ, c, c0 )-component is (respectively) F (ϕ, c0 ) and F (c, ϕ).
It is useful to stress that this characterization is compatible with thedescription of a co/limit as a co/equalizer, when F is mute in one variable.
From this we deduce a different argument showing that the co/end of a mutefunctor coincides with its co/limit.
Definition 1.16 : There is an obvious definition of preservation of co/ends
from their description as co/limits, which reduces to the preservation of theparticular kind of co/limit involved in the definition of end(F ) and coend(F ).
This remark entails easily that
Theorem 1.17 : Every co/continuous functor F : D → E preserves every´ /cco/end that exists in D, namely if T : Cop × C → D has a co/end c T (c, c),
then
´
 ´
/c
/c
F
T (c, c) ∼F T (c, c)
(17)
=
c

c

in the obvious meaning that the two objects are canonically isomorphic
having the same universal property.
As a particular example of this, we have

Corollary 1.18 [The hom functor commutes with integrals]: Continuity of the hom bifunctor C( , ) : Cop × C → Sets gives its co/end
preservation properties: for every c ∈ C we have the canonical isomorphisms
ˆ x
 ˆ
C
F (x, x), c ∼= C(F (x, x), c)
x
ˆ
 ˆ

C c, F (x, x) ∼= C(c, F (x, x))xx

The power of this remark can’t be overestimated: co/continuity of the hom
functor is a fundamental kata of coend-fu. Basically every example in the restof the paper involves a computation carried on using this co/end preservation
property, plus the fully faithfulness of the Yoneda embedding.

16

F. LOREGIAN

1.3. Natural transformations as ends. A basic example exploiting
the whole machinery introduced so far is the proof that the set of naturaltransformations between two functors F, G : C → D can be characterized as
an end:Theorem 1.19 : Given functors F, G : C → D between small categories wehave the canonical isomorphism of setsˆNat(F, G) ∼(18)
= D(F c, Gc).
c

Proof. Giving a wedge τc : Y → D(F c, Gc) consists in giving a function
y 7→ τc,y : F c → Gc, which is natural in c ∈ C (this is simply a rephrasing of
the wedge condition):G(f ) ◦ τc,y = τc0 ,y ◦ F (f )(19)

for any f : c → c0 ; this means that there exists a unique way to close thediagramτc/ D(F c, Gc)
O

Y

h

(

(20)

Nat(F, G)

Q

with a function sending y 7→ τ ,y ∈ c∈C D(F c, Gc), and where Nat(F, G) →D(F c, Gc) is the wedge sending a natural transformation to its c-component;
the diagram commutes for a single h : Y → Nat(F,´G), and this is precisely
the desired universal property for Nat(F, G) to be c D(F c, Gc).


Remark 1.20 : A suggestive way to express naturality as a “closure” conditionis given in [Yon60, 4.1.1], where for an Ab-enriched functor F : Cop ×C → Ab
between suitably complete Ab-categories, one can prove that Nat(F, G) =ker δ, for a suitable map δ defined among Ab(F x, Gx) ⊕ Ab(F y, Gy) andAb(F y, Gy).

After this, we can embark in more sophisticated and pervasive examples.
In particular, the following section is the gist of the paper, demonstrating
the power of co/end calculus to prove highly technical and involved resultsby means of abstract-nonsense only.

ENDS

17

Exercises for §1
E1 Show that for each f ∈ Sets(B, B 0 ) the following square is commutative:
0

XB × B0

X B ×f0

X f ×B

X B × B0

/ XB × B



(21)


/X

E2 A dinatural transformation serves to define natural transformations between
functors having the same co/domain but different variance; try to do this.
..
..
E3 Show with an example that dinatural transformations α : P −→ Q, β : Q −→Rcannot be composed in general. Nevertheless, there exists a “composition” of a..
..
dinatural α : P −→ Q with a natural η : P 0 → P which is again dinatural P 0 −
→ Q,
..
0
as well as a composition P −→ Q → Q (hint: the appropriate diagram results as
the pasting of a dinaturality hexagon and two naturality squares).
E4 Coends are obtained by a suitable dualization. State the definition of a cowedge
for a functor F : Cop × C → D; a coend for F consists of a universal cowedge
coend(F ) for P . Prove functoriality for coends. Show that coends are ends inthe dual category: the coend of F is the end of F op : C × Cop ∼= Cop × C → Dop .
E5 Define a category wd(F ) having objects the wedges for F : Cop × C → D andshow that the end of F is the terminal object of wd(F ); dualize to coends (initial
objects of a category cwd(F ) of cowedges).
E6 Show that extranatural transformations compose accordingly to these rules:• (stalactites) Let F, G be functors of the form Cop ×C → D. If αx,y : F (x, y) →G(x, y) is natural in x, y and βx : G(x, x) → H is extranatural in x (forsome object H of D), thenβx ◦ αx,x : F (x, x) → H(22)is extranatural in x.
• (stalagmites) Let G, H be functors of the form Cop × C → D. If αx : F →G(x, x) is extranatural in x (for some object F of D) and βx,y : G(x, y) →H(x, y) is natural in x, y, thenβx,x ◦ αx : F → H(x, x)(23)

is extranatural in x.
• (yanking) Let F, H be functors of the form C → D, and let G : C×Cop ×C →D be a functor. If αx,y : F (y) → G(x, x, y) is natural in y and extranaturalin x, and if βx,y : G(x, y, y) → H(x) is natural in x and extranatural in y,then
βx,x ◦ αx,x : F (x) → H(x)(24)
is natural in x.
Express these laws as equalities between suitable string diagrams (explaining
also the genesis of the names “stalactite” and “stalagmite”).
E7 Prove the Fubini theorem for ends embarking in a long exercise in universality;
prove that if F : Cop × C → D is mute in one of the two variables (i.e. F (c0 , c) =F̄ (c) or F̂ (c0 ) for each c, c0 ∈ C and suitable functors F̄ : C → D or F̂ : Cop → D),then the co/end of F is canonically isomorphic to its co/limit.

18

F. LOREGIANThis gives an alternative proof of a similar Fubini rule for co/limits: given afunctor F : I × J → D we have
∼ lim lim F =∼ limlimI limJ F =F(25)−→J −→I−→I×J−→ −→(and similarly for limits).
E8 Introduced to vector analysis in basic calculus courses, students learn that if~ : X → Rn such
(X, Ω, µ) is a measure space, the integral of a vector function Fthat each Fi = πi ◦F : X → R is measurable
 and has finite integral, is the vector´´
whose entries are
F
dµ,
·
·
·
,
F
dµ
.
X 1
X n
Prove that category theory possesses a similar formula, i.e. that if F : Cop × C →A1 × · · · × An is a functor towards a product of categories, such that for 1 ≤ i ≤ n• each Ai has both an initial and a terminal object, respectively denoted ∅and 1;´ /c• each co/end c πi ◦ F exists
´
´ /c
/c
then the “vector” of all these co/ends, as an object
F1 , . . . , c Fn ) ∈c

A1 × · · · × An , is the (base of a universal co/wedge forming the) co/end of F .
E9 The categories tw(C) and C§ are linked by a final (see [Bor94a, 2.11.1]) functor
K : C§ → tw(C); this motivates the fact that the colimit is the same whenindexed by one of the two. Define K and show that is is final, i.e. that for every
object in tw(C) the comma category (ϕ ↓ K) is nonempty and connected (see[Rie14, Remark 7.2.10, Example 8.3.9]).
E10 Show that the end of a functor T : ∆[1]op × ∆[1] → Sets is the pullback of theT (0,d0 )T (d0 ,1)morphisms T (0, 0) −−−−−→ T (0, 1) ←−−−−− T (1, 1), i.e.
´
/ T (0, 0)
T (i, i)i∈∆[1]T (1, 1)(26)

/ T (0, 1)

is a pullback in Sets (of course there’s nothing special about sets here!).
E11 Let G be a topological group, and Sub(G) the partially ordered (with respect toinclusion) set of its subgroups; let X be a G-space, i.e. a topological space witha continuous action G × X → X.
We can define two functors Sub(G) → Top, sending (H ≤ G) 7→ G/H (this is acovariant functor, and G/H has the induced quotient topology) and (H ≤ G) 7→
X H (the subset of H-fixed points for the action; this is a contravariant functor).
(1) Compute the coendˆ H≤G
X G × G/H
(27)
in Top if G = Z/2 has the discrete topology;´ H≤G G(2) Give a general rule for
X × G/H when H is cyclic with n elements;(3) Let instead Orb(G) be the orbit category of subgroups of G, whose objects
are subgroups but hom(H, K) contains G-equivariant maps G/H → G/K.
Let again X and G/ define the same functors, now with different

ENDS

19

´ H∈Orb(G) H
X × G/H ∼action on arrows. Prove that
= X (Elmendorf
reconstruction, [Elm83]).
(4) Let E|F be a field extension, and {H ⊆ Gal(E|F )} the partially ordered
set of subgroups of the Galois group of the extension. Compute (in the
category of rings)
ˆ H
E H × Gal(H|F )(28)
E12 Dualize
the above construction, to obtain a similar characterization for the coend´c
F (c, c), characterized as the coequalizer of a similar pair (F ∗ , F∗ ).
E13 Show that the hom functors C( , y) jointly preserve ends (see [ML98] for a
precise definition of joint preservation); the rough idea is that whenever fC(W, y)is the end of C(F´ c (c, c), y) for each y, naturally in y ∈ C, then W is canonicallyisomorphic to
F (c, c).
´
E14 Define the map δ above, and show that ker δ ∼= y Ab(F y, Gy) in the above
notation; dualize to express a coend as a suitable coequalizer.
´
E15 Prove again Theorem Q
1.19, using the characterization of c D(F c, Gc) as anequalizer: the subset of c∈C D(F c, Gc) you will have to consider is precisely thesubset of natural transformations {τc : F c → Gc | Gf ◦τc = τc0 ◦F f, ∀f : c → c0 }.
This yields the evocative formulaˆ
Nat(1C , 1C ) = End(1C ) ∼(29)= C(c, c).
c´c
Is it possible to give an explicit meaning to the dual construction
C(c, c) (startwith simple examples: C discrete, C a finite group, C a finite groupoid. . . )?
Compare also Example 2.11.
E16 What is the co/end of the identity functor 1Cop ×C : Cop × C → Cop × C? Use
the bare definition; use the characterization of co/ends as co/limits; feel free toinvoke Exercise E8.
E17 A preliminary definition for this exercise is the following: a set of objects S ⊂ C
finitely generates a category C if for each object X ∈ C and each arrow f : s → cfrom s ∈ S there is a factorization
g

s−
→

n
a

h

c
si −→
c

(30)

i=1

where hc is an epimorphism and {s1 , . . . , sn } ⊂ S (n depends on c and f ).
Suppose T : Cop × C → Sets is a functor, finitely continuous in both variables,
and C is finitely generated by S. Then, considering S as a full subcategory
S ⊆ C, and calling T 0 := T |S : Sop × S → Sets, we have an isomorphism
ˆ c∈S
ˆ c∈C
∼
T 0 (c, c) =T (c, c)(31)
´ c∈S 0´ c∈Cinduced by a canonical arrow
T (c, c) →T (c, c).
E18 Let F a U : C  D be an adjunction, and G : Dop × C → E a functor; then there
is an isomorphismˆˆc

G(F c, c) ∼=d

G(d, U d).

(32)

Show that a converse of this result is true: if the above isomorphism is true for
any G and natural therein, then there is an adjunction F a U .

20

F. LOREGIAN

2. Yoneda reduction, Kan extensions
One of the most famous results about the category Cat(Cop , Sets) ofpresheaves on a category C is that every object in it can be canonicallypresented as a colimit of representable functors; see [ML98, Theorem III.7.1]
for a description of this classical result (and its dual holding in Cat(C, Sets)).
Now, co/end calculus allows us to rephrase this result in an extremelycompact way, called Yoneda reduction; in a few words, it says that every
co/presheaf can be expressed as a co/end 4.
Proposition 2.1 [ninja Yoneda Lemma]: For every functor K : Cop → Sets
and H : C → Sets, we have the following isomorphisms (natural equivalencesof functors):
ˆ c
ˆ
∼
∼
(ii) K = KcC(c, )(i) K =
Kc × C( , c)ˆ cˆc(iii) H ∼Hc × C(c, )(iv) H ∼== HcC( ,c)cRemark 2.2 : The name ninja Yoneda lemma is a pun coming from a
Math-Overflow comment by T. Leinster, whose content is basically theproof of the above statement:Th[e above one is] often called the Density Formula, [. . . ] or (byAustralian ninja category theorists) simply the Yoneda Lemma.
(but Australian ninja category theorists call everything the Yoneda
Lemma. . . ).

Undoubtedly, there is a link between the above result and the Yoneda
Lemma we all know: in fact, the proof heavily relies on the Yoneda isomorphism, and in enriched setting (see [Dub70, §I.5]) the ninja Yoneda lemma,interpreted as a theorem about Kan extensions, is equivalent to the one fromthe Northern hemisphere.
We must admit to feel somewhat unqualified to properly discuss the topic,
as we live in the wrong hemisphere of the planet to claim any authority
on it, and to be acquainted with the rather unique taste of Australianpracticioners in choosing evocative (or obscure) terminology. Nevertheless,along the whole note, we keep the name “ninja Yoneda Lemma” as a(somewhat witty) nickname for the above isomorphisms, without pretending
any authoritativeness whatsoever.
4The reader looking for a nifty explanation of this result should wait for a more thoroughdiscussion, which can be deduced from the material in §4, thanks to the machinery ofweighted co/limits.

ENDS

21

Proof. We prove case (i) only, all the others being totally analogous. We
put a certain emphasis on the style of this proof, as it is paradigmatic ofmost of the subsequent ones. Consider the chain of isomorphisms
´
 ˆ

c∈C
Sets
Kc × C(x, c), y ∼Sets Kc × C(x, c), y=ˆc∈C∼
Sets(C(x, c), Sets(Kc, y))=c∈C∼ Nat(C(x, ), Sets(K−, y))=∼= Sets(Kx, y)

where the first step is motivated by the coend-preservation property of thehom functor, the second follows from the fact that Sets is a cartesian closed
category, whereSets(X × Y, Z) ∼(33)
= Sets(X, Sets(Y, Z))

for all three sets X, Y, Z (naturally in all arguments), and the final step
exploits Theorem 1.19 plus the classical Yoneda Lemma.
Every step of this chain of isomorphisms is natural in y; now we have only
to notice that the natural isomorphism of functors
´

c
Sets
Kc × C(x, c), y ∼(34)= Sets(Kx, y)´c
Kc × C(x, c) ∼ensures that there exists a (natural) isomorphism= Kx.
This concludes the proof and it is, if you want, a way to motivate andpartially solve Exercise 1.E13.

From now one we will make frequent use of the notion of (Sets-)tensor and(Sets-)cotensor in a category; these standard definitions are in the chapterof any book about enriched category theory (see for example [Bor94b, Ch.
6], its references, and in particular its Definition 6.5.1, which we report for
the ease of the reader:
Definition 2.3 [Tensor and cotensor in a V-category]: In any Venriched category C (see [Bor94b, Def. 6.2.1]), the tensor · : V × C → C isa functor (V, c) 7→ V · c such that there is the isomorphism
C(V · c, c0 ) ∼(35)= V(V, C(c, c0 )),natural in all components; dually, the cotensor in an enriched category C is afunctor (V, c) 7→ cV (contravariant in V ) such that there is the isomorphism
C(c0 , cV ) ∼(36)= V(V, C(c0 , c)),natural in all components.

Example 2.4 : Every co/complete,
locally small category
C is naturally SetsQ`∼
co/tensored by choosing cV ∼
c
and
V
·
c
= v∈V
= v∈V c.

22

F. LOREGIAN

Remark 2.5 : The tensor, hom and cotensor functors are the prototype ofa thc situation (see our Remark 3.13 and [Gra80, §1.1] for a definition);
given the hom-objects of a V-category C, the tensor · : V × C → C andthe cotensor ( )( ) : Vop × C → C can be characterized as adjoint functors:
usual co/continuity properties of the co/tensor functors are implicitly derived
from this characterization.
Remark 2.6 [ , The Yoneda embedding is a Dirac delta]: In functional analysis, the Dirac delta appears in the following convenient abuse ofnotation:ˆ ∞f (x)δ(x − y)dx = f (y)(37)−∞(the integral sign, here, is not a co/end). Here δy (x) := δ(x − y) is the
y-centered delta-distribution, and f : R → R is a continuous, compactly
supported function on R.
It is really tempting to draw a parallel between this relation and theninja Yoneda lemma, conveying the intuition that representable functors
on an object c ∈ C play the rôle of c-centered delta-distributions. If therelation above is written as hf, δy i = f (y), interpreting integration as an
inner product between functions, then the ninja Yoneda lemma says formallythe same thing:for each presheaf F : Cop → Sets, the “inner product”´x
hよc , F i =よc (x) × F c equals F c (obviously, there’s nothing special
about sets here).
2.1. Kan extensions as co/ends.
Definition 2.7 : Given a functor F : C → D, its left and right Kan extensions
are defined5 to be, respectively, the left and right adjoint to the “precomposition” functor
F ∗ : Fun(D, E) → Fun(C, E)(38)∗
given by H 7→ F (H) = H ◦F , in such a way that there are two isomorphisms∼ Nat(G, H ◦ F )Nat(LanF G, H) =Nat(H ◦ F, G) ∼= Nat(F, RanF G).

Now, we want to show that in “nice” situations it is possible to describe
Kan extensions via co/ends: whenever the co/tensors (see Def. 2.3) involvedin the definition of the following co/ends exist in D for any choice of functors
F, G (a blatant example is when D is co/complete, since in that case as said
5This is not true, strictly speaking. Nevertheless we prefer to cheat the reader with thisuseful insight instead of obscuring the general idea keeping track of all possible pathologies.
Furthermore, we are interested only in the cases when Kan extensions can be written as
co/ends, so we will not consider any pathology whatsoever and we can safely assume that
this is a proper definition.

ENDS

23

before it is always Sets-co/tensored by suitable iterated co/products, see theremark above 2.5), then the left/right Kan extensions of G : C → E along
F : C → D exists and there are isomorphisms
ˆ
ˆ c
∼
∼
RanF G = GcD( ,F c) .
(39)
LanF G =D(F c, ) · Gcc

Proof. The proof consists of a string of canonical isomorphisms, exploiting
simple remarks in elementary Category Theory and the results establishedso far: the same argument is offered in [ML98, Thm. X.4.1, 2].
Nat

´

c

D(F c, ) · Gc, H

 1.19 ´

´



∼
=

xD

1.18 ´cx D(D(F c, x) · Gc, Hx)∼=

cx Sets(D(F c, x), E(Gc, Hx))

1.19 ´∼=c Nat(D(F c,∼=(35) ´

Yon ´

∼
=

c

D(F c, x) · Gc, Hx), E(Gc, H−))c E(Gc, HF c) = Nat(G, HF ).

The case of RanF G is dually analogous.

∼Remark 2.8 : This is the pattern of every “proof by coend-juggling” we willmeet in the rest of the paper; from now on we feel free to abandon a certain
pedantry in justifying every single deduction in the chains of isomorphisms
leading to conclude a proof.
Proposition 2.9 : Left/right adjoint functors commute with left/right Kan
extensions, whenever they can be expressed as the coends above.
Proof. An immediate corollary of Theorem 1.17, once it has been provedthat a left adjoint commutes with tensors, i.e. F (X · a) ∼= X · F a for any(X, a) ∈ Sets × C.
Example 2.10 : Let T : C → C be a monad on the category C; the Kleisli
category Kl(T ) of T is defined having the same objects of C and morphisms
Kl(T )(a, b) := C(a, T b).
Given any functor F : A → C, the right Kan extension TF = RanF F isa monad on C, the codensity monad of F ; hom-sets in the Kleisli category
Kl(TF ) can be characterized asˆ
0 ∼Kl(TF )(c, c ) = Sets(C(c0 , F a), C(c, F a)).
(40)a(39) ´The proof is an exercise in coend-juggling, recalling that TF ( ) ∼=C( ,F c) .
c Fc

24

F. LOREGIAN

Example 2.11 : Let V be a finite dimensional vector space over the field K;
let V ∨ denote the dual vector space of linear maps V → K. Then there is a
canonical isomorphism
ˆ V
V∨⊗V ∼(41)= K.

The fastest way to see this is to notice that
ˆ V
∼ Lanid (id) ∼hom(V, ) ⊗ V == idVect(42)(compare this argument with any proof trying to explicitly evaluate thecoend!)V
Remark 2.12 : The universal cowedge hom(V, V ) −−→
K sends an endomorphism f : V → V to its trace τ (f ) ∈ K (which in this way acquires a universal
property).
The above argument holds in fact in fair generality, adapting to the casewhere V is an object of a compact closed monoidal category C.

αExercises for §2
E1 Show that presheaf categories are cartesian closed, via coends: if [Cop , Sets] isthe category of presheaves on a small C, then there exists an adjunction
Q
Nat(P × Q, R) ∼(43)
= Nat(P, R )by showing that RQ (c) = Nat(よc × Q, R) does the job (use the ninja Yoneda
lemma, as well as Thm. 1.19).
E2 Use equations (39) and the ninja Yoneda lemma that Lanid and Ranid are theidentity functors, as expected. Use again (39) and the ninja Yoneda lemma
to complete the proof that F 7→ LanF is a pseudofunctor, by showing that forFG
HA−→ B, A −→ C −→ D there is a uniquely determined laxity cell for composition∼ LanHG (F )LanH (LanG (F )) =
(44)
´ xy
(hint: coend-juggle with LanH (LanG (F ))d until you get(D(Hx, d)×C(Gy, x))·F y; now use the ninja Yoneda lemma plus co-continuity of the tensor, as suggested in Remark 2.5).
E3 Prove in a similar way
 isomorphisms (ii), (iii), (iv) in Thm. 2.1 (hint: for (ii) and´C(x,c)(iv) start from Sets y, c Hcand use again the end preservation property,
cartesian closure of Sets, and Thm. 1.19).
E4 Let C´ be a compact closed monoidal category [Day77]; Show that the functorx ∨y 7→x ⊗ y ⊗ x carries the structure of a monad on C.

3. The nerve and realization paradigm.
3.1. The classical nerve and realization. The most fruitful application of the machinery of Kan extensions is the “Kan construction” for the

ENDS

25

realization of simplicial sets. It is impossible to underestimate the valueof this construction as a unification tool in algebra and algebraic topology;
here, we briefly sketch this construction.
Consider the category ∆ of finite ordinals and monotone maps, as definedin [GJ09], and the Yoneda embedding よ∆ : ∆ → sSet; we can define twofunctors ρ : ∆ → Top and i : ∆ → Cat which “represent” every object [n] ∈ ∆either as a topological space or as a small category:• The category i[n] is {0 → 1 → · · · → n} (there is a similar functor
P : Pos → Cat regarding any poset (P, ≤) as the category P where
the composition function is induced by the partial order relation ≤:i here is the restriction P |∆ : ∆ ⊂ Pos);• The topological space ρ[n] is defined as the standard n-simplex ∆nembedded in Rn+1 ,nρ[n] = (x0 , . . . , xn ) ∈ Rn+1 | 0 ≤ xi ≤ 1,Pnoi=0 xi = 1

.

(45)

In a few words, we are in the situation depicted by the following diagrams:
∆
よ

i

∆

/ Cat

よ



sSet

ρ

/ Top

(46)



sSet

The two functors i, ρ can be (left) Kan-extended along the Yoneda embeddingよ∆ : ∆ → sSet, and these extensions happen to be left adjoints (this canbe proved directly, but we will present in a while a completely generalstatement).
We denote these adjunctions
Lanよ i a Ni and Lanよ ρ a Nρ ;(47)

these two right adjoint functors are called the nerves associated to i andρ respectively, and are defined, respectively, sending a category C to thesimplicial set Ni (C) : [n] 7→ Fun(i[n], C) (the classical nerve of a category),
and to the simplicial set Nρ (X) : [n] 7→ Top(ρ[n], X) = Top(∆n , X) (thesingular complex of a space X).6
The left adjoints to Nρ and Ni must be thought as “realizations” of asimplicial set as an object of Top or Cat:
6The name is motivated by the fact that if we consider the free-abelian group onNρ (X)n , the various Cn = Z · Nρ (X)n =`Nρ (X)nhomology is precisely the singular homology of X.

Z organize as a chain complex, whose

26

F. LOREGIAN• The left Kan extension Lanよ ρ is the “geometric” realization |X• | ofa simplicial set X• , resulting as the coendˆ n∈∆∆∆n × Xn(48)
which is equal to a suitable coequalizer in Top. The shape of thisobject is fairly easy to motivate, keeping open any book in algebraic topology. The topological space |X• | is obtained choosing an-dimensional disk ∆n (we are, of course, reasoning up to isomorphism!) for each n-simplex x ∈ Xn and gluing these disks along
their boundaries δi (∆n ) according to the degeneracy maps of X• .
The resulting space is (almost by definition) a cw-complex (thismeans that |X• | has the topology induced by a sequential colimit of
pushouts of spaces X(0) → X(1) → . . . ).
• The left Kan extensionLanよ i is the “categorical realization” τ1 (X• )of a simplicial set X• , resulting as the coendˆ n∈∆∆i[n] × Xn .
(49)
This is the category whose objects are 0-simplices of X• , arrows are
1-simplices, and the map X1 ×X0 X1 → X1 obtained pulling backd0 , d1 : X1 → X0 works as a composition law. The category i[n]can be regarded as the “universal composable string of n arrows”,
and the action of degeneracies (given by composition of contiguous
arrows) induces a composition law for n-tuples of arrows under thequotient that defines the coend.
We leave the reader think about why we only mention degeneracies here,leaving out faces (they are necessary, aren’t they?), and we depict thegeometric realization of X• in figure 3.1.
It should now be evident that there is a pattern (we will call it the “nerverealization paradigm”) acting behind the scenes, and yielding the classical/singular nerve as particular cases of a general construction interpreted
from time to time in different settings; unraveling this machinery with thepower of co/end calculus is the scope of the following section.
3.2. Some famous realizations and their associated nerves. Algebraic topology, representation theory, and more generally every setting wherea “well-behaved” categorical structure is involved constitute natural factories
for examples of the nerve-realization paradigm. We now want to lay downorganic foundations for a general theory and a general terminology allowing
us to collect several examples (leaving outside many interesting others!) of

ENDS

27

Figure 1. The construction of the geometric realization ofX ∈ sSet as a sequential colimit of pushouts of disks.
nerve-realizations pairs, obtained varying the domain category of “geometricshapes” or the category where this fundamental shapes are “represented”.
Definition 3.1 : Any functor ϕ : C → D from a small category C to a (locallysmall) cocomplete category D is called a nerve-realization context (a nrcontext for short).
Given a nerve-realization context ϕ, we can prove the following result:
Proposition 3.2 [Nerve-realization paradigm]: The left Kan extensionof ϕ along the Yoneda embedding よ : C → [Cop , Sets], i.e. the functor
Rϕ = Lanよ ϕ : [Cop , Sets] → D is a left adjoint, Rϕ a Nϕ . Rϕ is called theD-realization functor or the Yoneda extension of ϕ, and its right adjoint the
D-coherent nerve.
Proof. The cocomplete category D is Sets-tensored, and hence Lanよ ϕ canbe written as the coend in equation (39); so the claim follows from the chainof isomorphisms
ˆ cD Lanよ ϕ(P ), d ∼[Cop , Sets](よc , P ) · ϕc, d=Dˆ∼= D [Cop , Sets](よc , P ) · ϕc, dˆc∼= Sets [Cop , Sets](よc , P ), D(ϕc, d)ˆc∼= Sets P c, D(ϕc, d) .
c

28

F. LOREGIAN

If we define Nϕ (d) to be c 7→ D(ϕc, d), this last set becomes canonicallyisomorphic to [Cop , Sets](P, Nϕ (d)).
Remark 3.3 : The nerve-realization paradigm can be rewritten in the following equivalent form: there is an equivalence of categories, induced by theuniversal property of the Yoneda embedding,b D)
Fun(C, D) ∼(50)
= RAdj(C,whenever D is a cocomplete locally small category (in such a way that“the category of nerve-realization contexts” is a high-sounding name for the
category of functors Fun(C, D)).

A famous result in Algebraic Topology (see for example [GZ67, GJ09])
says that the geometric realization functor R : sSet → Top commutes withfinite products: coend calculus gives a massive simplification of this result.
Proof. The main point of the proof is showing that the geometric realization
commutes with products of representables: the rest of the proof relies on asuitable application of coend-fu. We could appeal conceptual ways to show
this preliminary result ([ABLR02, §2]):
Proposition 3.4 : The following properties for a functor F : C → Sets areequivalent:• F commutes with finite limits;• Lanよ F commutes with finite limits;• F is a filtered colimit of representablefunctors;´• The category of elements C F of F (see Def. 4.1 and Prop. 4.14) iscofiltered.
Nevertheless, this result isn’t powerful enough to show that the geometricrealization ρ commutes with Top-products, since it gives only a bijection|∆[n] × ∆[m]| ∼= ∆n × ∆m ; a certain amount of dirty work is necessary
to show that this bijection is a homeomorphism. However, if we take thecommutativity of R with finite products of representables for granted, theproof is a kata in coend-fu, recalling that
• The geometric realization is a left adjoint, hence it commutes withcolimits and tensors;• Every simplicial set is a colimit of representables.
Starting the usual machinery, we have that
´n
 ´m


R(X × Y ) ∼Xm · ∆[m] ×Yn · ∆[n]
=R
´ mn

∼
(Xm · ∆[m]) × (Yn · ∆[n])
=R
´ mn

∼
(Xm × Yn ) · (∆[m] × ∆[n])
=R
´ mn
∼
(Xm × Yn ) · R(∆[m] × ∆[n])
=

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29

´ mn
∼
(Xm × Yn ) · ∆m × ∆n=
´ mn∼
(Xm · ∆m ) × (Yn · ∆n )
=
´m
´n


∼
Xm · ∆m ×
Yn · ∆n=∼= R(X) × R(Y )where we applied, respectively, the ninja Yoneda lemma, the colimit preservation property of R, its commutation with tensors, and its commutativity
with finite products of representables.

3.3. Examples of nerves and realizations. A natural factory of nerverealization contexts is homotopical algebra, as such functors are often used
to build Quillen equivalences between model categories. This is somewhat
related to the fact that “transfer theorem” for model structures often apply
to the well-behaved nerve functor.
But Quillen adjunctions between model categories are certainly not the
only examples of nr-paradigms!
The following list attempts to gather important examples of nr-contexts:
for the sake of completeness, we repeat the description of the two abovementioned examples of the topological and categorical realizations.
Example 3.5 [Categorical nerve and realization]: In the case of ϕ =i : ∆ → Cat, we obtain the classical nerve NCat of a (small) category C,
whose left adjoint is the categorical realization (the fundamental category
τ1 X of X described in [Joy02b]). The nerve-realization adjunction
τ1 : sSet  Cat : NCat(51)

gives a Quillen adjunction between the Joyal model structure on sSet (see[Joy02b]) and the folk model structure on Cat.
Example 3.6 [Geometric nerve and realization]: If ϕ = ρ : ∆ → Top is
the realization of a representable [n] in the standard topological simplex, we
obtain the adjunction between the geometric realization |X| of a simplicial
set X and the singular complex of a topological space Y , i.e. the simplicial
set having as set of n-simplices the continuous functions ∆n → Y .

Example 3.7 [sSet-coherent nerve and realization]: If ϕ : ∆ → Cat∆ isthe functor which realizes every representable [n] as a simplicial category having objects the same set [n] = {0, 1, . . . , n} and as hom(i, j) the simplicial set
obtained as the nerve of the poset P (i, j) of subsets of the interval [i, j] whichcontain both i and j,7 we obtain the (Cordier) simplicially coherent nerve
and realization, which sends C into a simplicial set constructed “coherently
remembering” that C is a simplicial category. This adjunction establishes a
7In particular if i > j then P (i, j) is empty and hence so is its nerve.

30

F. LOREGIAN

Quillen adjunction sSet  Cat∆ which restricts to an equivalence between
quasicategories (fibrant objects in the Joyal model structure on sSet) andfibrant simplicial categories (with respect to the Bergner model structure onCat∆ ).
Example 3.8 [Moerdijk generalized intervals]: The construction giving
the topological realization of ∆[n] extends to the case of any “interval” inthe sense of [Moe95, §III.1], i.e. any ordered topological space J having“endpoints” 0, 1; indeed every such space J defines a “generalized” (in thesense of [Moe95, §III.1]) topological n-simplex ∆n (J), i.e. a nerve-realization
context ϕJ : ∆ → Top.

Example 3.9 [Toposophic nerve and realization]: The correspondenceδ : [n] 7→ Sh(∆n ) defines a cosimplicial topos, i.e. a cosimplicial object in
the category of toposes, which serves as a nr-context. Some geometricproperties of this nerve/realization are studied in [Moe95, §III]: we outline
an instance of a problem where this adjunction naturally arises: let X, Y be
the categories of sheaves over topological spaces X, Y . Let X ? Y be the joinof the two toposes seen as categories: this blatantly fails to be a topos, butthere is a rather canonical “replacement” procedure
Cat × Cat(X, Y)

?

Cat
X?Y

N

Lanよ (δ)

sSetCat(∆• , X ? Y)

Topos
X~Y

Example 3.10 [The Dold-Kan correspondence]: The well-known DoldKan correspondence, which asserts that there is an equivalence of categories∆op , Ab] and chain complexes Ch+ (Ab)between simplicial abelian groups [∆with no negative homology, and it can be seen as an instance of the nerverealization paradigm.
In this case, the functor ∆ → Ch+ (Ab) sending [n] to Z∆[n] (the freeabelian group on ∆[n]) and then to the Moore complex M (Z∆[n] ) determined∆op , Ab] as in [GJ09] is the nerve-realization
by any simplicial group A ∈ [∆context.
Example 3.11 [Étale spaces as Kan extensions]: Let X be a topologicalspace, and Opn(X) its poset of open subsets. There exists a natural functor
A : Opn(X) → Top/X(52)

Lanよ A a NA

(53)

sending U ⊆ X to the same morphism U → X; this works as a nr-context
giving the pair of adjoint functors
where NA is defined precisely taking the (pre)sheaf of sections of p ∈ Top/X .
The resulting left adjoint is precisely the functor sending a presheaf F ∈

ENDS

31

[Opn(X), Sets] to the space whose carrier is the disjoint union of stalks`
F̃ = x∈X Fx , endowed with the final topology turning all maps of the forms̃ : U → F̃ sending y to the equivalence class [s]y ∈ Fy .
This adjunction restricts to an equivalence of categories between thesubcategory Sh(X) of sheaves on X and the subcategory Ét(X) of étale
spaces over X, giving a formal method to prove [MLM92, Thm. II.6.2]. A
complete proof can be found at [Car], lectures 3 and 4.
Example 3.12 [The tensor product as a coend]: Any ring R can be
regarded as an Ab-category with a single object, whose set of endomorphisms
is the ring R itself; once noticed this, we obtain natural identifications forthe categories of modules over R:
ModR ∼= Fun(Rop , Ab)∼
R Mod = Fun(R, Ab).
Given A ∈ ModR , B ∈ R Mod, we can define a functor TAB : Rop × R → Ab
which sends the unique object to the tensor product A ⊗Z B of abeliangroups. The coend of this functor can be computed as the coequalizer
coker
or in other words,

`r∈R A ⊗Z B

ˆ ∗∈R

r⊗1
1⊗r


/
/ A ⊗Z B ,

(54)

TAB ∼
= A ⊗R B. This point of view on tensor

products can be extremely generalized (see [ML98, §IX.6], but more on thishas been written in [Yon60, §4]): given functors F, G : Cop , C → V having
values in a cocomplete monoidal category, we can define the tensor productof F, G as the coendˆ c
F C G :=F c ⊗ V Gc.
(55)

Remark 3.13 : This can be regarded as part of a general theory which definesa thc situation (see [Gra80, §1.1]; these are also called adjunctions of two
variables in newer references) as a triple t = (⊗, ∧, [ , ]) of (bi)functors
between three categories S, A, B, defined via the adjunctions
B(S ⊗ A, B) ∼= S(S, [A, B]) ∼= A(A, S ∧ B).

(56)

Such an isomorphism uniquely determines the domains ant the variance of thethree functors involved, in each variable; to be more clear, however, we noticethat ⊗ : S × A → B, and then ∧ : Sop × B → A, and [ , ] : Aop × B → S.

Example 3.14 [Giraud theorem using coends]: The gist of Giraud theorem is the following statement: left exact localizations of presheaf categories

32

F. LOREGIAN[Cop , Sets] classify Grothendieck toposes (i.e. categories of sheaves Sh(C, J)
with respect to a Grothendieck topology J).
A proof of this classical “representation” theorem, intertwined with thetheory of locally presentable categories (see for example [Vit06]), is containedat the end of [MLM92].
We now try to outline an argument giving the localization between apresheaf category and a category E satisfying the axioms of Giraud, hence“realizing” E as a full subcategory of [Cop , Sets]. presheaves on C = Ec ⊂ E,
the subcategory of compact object of E.
The trick in the proof is to choose C wisely: to do this we use the factthat there is a small full subcategory C ⊆ E of compact objects, E<ω , and afull embedding ι : C ⊂ E; this is a nerve-realization context (Def. 3.1), that
activates coend calculus to prove that
(1) The ι-nerve Nι is full and faithful and coincides with the inclusionof sheaves into presheaves [Cop , Sets];(2) Lanよ ι is the left exact reflection.
Let then
Lanよ (ι) : [Cop , Sets] o

/

E: Nι

(57)

be the nerve-realization adjunction.
Since the functor よC is dense, the associated nerve Nι is fully faithful,and this gives the first point: it remains then only to prove that the functor
Lanよ (ι) behaves like sheafification. This, in view of our characterization of
the unit and counit of the nerve-realization adjunction (Remark E4) meansthat we have to manipulate the following chain of (iso)morphisms:
Lanよ (ι)(P ) ∼=E(ιC, Lanよ ι(P ))´A
∼P A × ιA
=E ιC,´A←E ιC, P A × ιA´A
∼P A × E ιC, ιA=´A
∼P A × C C, A=∼
=P C
It only remains to prove that this functor is left exact. To do this we invoke
Prop. 3.4. It also remains to characterize sheaves as those P such that ηP
is invertible (this to a certain amount seems to be implied by taking only
those P that preserve finite limits, and yet. . . ).
Example 3.15 [Simplicial subdivision functor]: Let ∆ be the standard“topologist’s” simplex category. The Kan Ex∞ functor is an endofunctor of

ENDS

33

∆op , Sets] turning every simplicial set X into a Kan complex 8. ThissSet = [∆construction is of fundamental importance in simplicial homotopy theory,
and we now want to organize the classical construction in the modern termsof a nerve-realization paradigm on ∆ .
First of all, recall ([GJ09]) that the nondegenerate m-simplices of ∆[n]are in bijective correspondence with the subsets of {0, . . . , n} of cardinality
m + 1; this entails that the set of nondegenerate simplices of ∆[n] becomes aposet s[n] ordered by inclusion when this partial order is tacitly transported.
We can then consider the nerve Nρ (s[n]) ∈ sSet (see Example 3.6). This
organizes into a functor sd : ∆ → sSet, which forms a nerve-realization
paradigm: using Prop. 3.2 we obtain the adjunction
sSet

Ex

sSet

⊥

Sd

where Ex is the nerve Nsd associated to the nr-paradigm sd: the set ofm-simplices Ex(X)n is sSet(sd(∆[n]), X).
There is a canonical map sd(∆[n]) → ∆[n] which in turn, by the Yoneda
lemma, induces a map X → Ex(X), natural in X ∈ sSet. This gives toEx( ) the structure of a pointed functor, and in fact a well-pointed functor
in the sense of [Kel80]; this, finally, means that we can defineEx∞ (X) ∼X → Ex(X) → Ex2 (X) → · · ·
= lim
−→



(58)

as an endofunctor on sSet. The functor Ex∞ enjoys a great deal of formal
properties useful in the study of simpicial homotopy theory (the mostimportant of which is that Ex∞ (X) is a Kan complex for each X ∈ sSet, see[GJ09]). A more intrinsic characterization of this construction is containedin [EP08], and defines not only Sd = Lanよ sd is a Left Kan extension, butalso sd: they consider the diagram of 2-cells∆ ×∆⊕よ∆ ×よ∆

sSet × sSet

×

sSet

sd

∆
よ∆

sSet

Sd

where ⊕ : ∆ × ∆ → ∆ is the ordinal sum defined by [m] ⊕ [n] = [m + n + 1].
8A Kan complex is a simplicial set Y such that the functor hom( , X) turns each horn
inclusion Λk [n] → ∆[n] into an epimorphism.

34

F. LOREGIAN

Example 3.16 [Isbell duality]: Isbell duality consists of the following
statement: let V be a Bénabou cosmos, and C ∈ V-Cat; if we denote, as
always, [C, V] and [Cop , V] the categories of covariant and contravariant
functors C → V, then we have an adjunctionO[C, V]op⊥

Spec[Cop , V].

This means that we find a bijection of hom-sets∼ [Cop , V](X, Spec(Y ))[C, V]op O(X), Y = [C, V] Y, O(X) =induced by the functors

O: X 7→ c 7→ [C, V] X, よCop (c)Spec : Y 7→ c 7→ [C, V]op よCop (c), Y

(59)



Executed by an expert in coend-fu, this statement is almost a tautology
thanks to Thm. 1.19:
ˆ
ˆ



∼
[C, V] Y, O(X) =V Y d, V(Xa, C(a, d))aˆd∼V Y d, V(Xa, C(a, d))
=
ˆda 
ˆ

∼
V Xa, V(Y d, C(a, d))
=
a

d


∼
= [Cop , V] X, Spec(Y ) .

Exercises for §3
E1 Use coend-fu to show that starting from a given thc-situation t = (⊗, ∧, [ , ]),op
we can induce a new one t0 = (, f, h , i), on the categories SI ×J , AI , BJ , forop
any I, J ∈ Cat; start defining F  G ∈ BJ out of F ∈ SI ×J , G ∈ AI , as the coendˆ i
F (i, ) ⊗ Gi
(60)
and show that there is an adjunction
I∼ SIop ×J (F, hG, Hi) ∼BJ (F  G, H) == A (G, F f H)
J

(61)

developing B (F  G, H) = . . . in two ways.
E2 In this exercise Top is a nice category for algebraic topology. Define the category
Γ having objects the power-sets of finite sets, and morphisms the functions
f : 2n → 2m preserving unions and set-theoretical differences.
(1) Show that there is a functor ∆ ,→ Γ, sending the chain {0 < 1 < · · · < n}in ∆ to {∅ ⊂ {0} ⊂ · · · ⊂ {0, . . . , n}} in Γ.

ENDS

35

(2) The category of presheaves of spaces Γop → Top is called the category ofΓ-spaces; a Γ-space is Segal if it turns pushout in Γ (describe them) into
homotopy pullback in Top.
More explicitly, let A : Γop → Top be a Γ-space,
Qn it is Segal if (a) A(0)is contractible; (b) the canonical map A(n) → i=1 A(1) is a homotopy
equivalence in Top.
(3) Let X ∈ Top and A : Γop → Top; define X ⊗ A to be the coend (in Top)ˆ n∈ΓX n × A(n)(62)
Show that S 1 ⊗Γ A is homeomorphic to the geometric realization of theA→ Top. If A is Segal, S 1 ⊗Γ A ∼simplicial space ∆ op → Γop −= BA(1), whereB(−) is the classifying space functor.
(4) Let C : Γop → . . . ; let X ⊗Γ C be the coend (in the category of topologicalcategories)
ˆ n∈Γ
X n × C(n).
(63)

Show that X ⊗Γ ( ) : Top-Cat → Cat commutes with finite products,
namely if C, D are topological categories, then∼ (X ⊗Γ C) × (X ⊗Γ D).
X ⊗Γ (C × D) =(64)

E3 Compute the J-realization (see Example 3.8) of X ∈ sSet in the case J is the
Sierpiński space {0 < 1} with topology {∅, J, {1}}.
E4 Write explicitly the unit and counit of the nerve and realization adjunction
Lanよ ϕ a Nϕ .
E5 Show that the nerve functor Nϕ is canonically isomorphic to Lanϕ よ, so that
there is an adjunction
Lanよ ϕ a Lanϕ よ.
(65)
E6 [ ] Example 3.9 can be expanded and studied more deeply:• Is a monoidal structure on Topos?
• Under which conditions on X, Y is X Y equivalent to a topos of sheaves
on a topological space X Y ?
• What are the properties of the bifunctor (X, Y ) 7→ X Y ? Does thisoperation resemble or extend the topological join?
E7 Generalize the nerve-realization paradigm to the setting of separately cocontinuous, or multilinear functors. Given ϕ : C1 × · · · × Cn → D, where each Ci issmall and D is cocomplete,show that there exists an equivalence of categories

c1 × · · · × C
cn , D)
Cat(C1 × · · · × Cn , D) ∼= Mult(C(66)

where Mult( , ) is the category of all functors that are cocontinuous in eachvariable once all the others have been fixed (hint: show it ‘by induction’ composing
multiple Kan extensions). Given θ ∈ Cat(C1 × · · · × Cn , D), describe the rightadjoint of each θ(c1 , . . . , i , . . . , cn ) : Cbi → D. All these functors assemble to ac1 × · · · × C
cn .
‘vector-nerve’ N : D → C

4. Weighted limits.

The theory of weighted co/limits constitutes an extremely interestingargument even per se and it constitutes one of the pillars of enriched category

36

F. LOREGIAN

theory. It can be easily formulated and understood in terms of co/end
calculus.
The whole discussion in this chapter comes from [Rie14, II.7]; even mostof the notation is basically the same. A classical reference for the subject
is [Kel82, KS74]: we do not make any attempt to recall the fundamental
definitions of enriched category theory, heavily relying on these classicalreferences.
It’s easy to motivate that the calculus of co/limits is a cornerstone ofbasic category theory. Nevertheless, this notion becomes too strict when one
deals with enriched categories; the “conical” shape of a classical co/limit
is not general enough to encompass the fairly rich variety of shapes inwhich co/limits in enriched categories arise; naïvely speaking, one of theseshapes gives rise to the notion of a cone, having as limit the well-known
initial/terminal object. In full generality, one can compute a co/limit of adiagram F : J → C weighing it with a presheaf with the same domain of F :conical colimits arise weighing with the terminal presheaf on Sets.
We cannot touch but the surface of this intricate topic: the interested
reader can consult [Kel89], a presentation of unmatched lucidity filled withenlightening examples. We choose to follow a more modern approach, for
we are interested in translating the fundamentals about weighted co/limits
into a kata of co/end-fu. To do this, we follow another really useful andwell-written presentation of the theory, c’est à dire [Rie14].9
4.1. A brief prelude: the category of elements of a presheaf.
Definition
4.1 : Let W : C → Sets be a functor; the category of elements´
C W of W is the category having objects the pairs (c ∈ C, u ∈ W c), andmorphisms (c, u) → (c0 , v) those f ∈ C(c, c0 ) such that W (f )(u) = v.
´
Notation 4.2 : The exotic notation “C W ” for the category of elements of
W comes from the wondrous paper [Gra].
´
Proposition 4.3 : The category C W can be equivalently characterized as• The category which results from the pullback´/ Sets∗
C W
(67)


C

y

W



U

/ Setswhere U : Sets∗ → Sets is the forgetful functor which sends a pointed
set to its underlying set;
9The reader is warned, now, that the present section results in a rough imitation ofthis source. Every error is obviously due to the author’s misunderstanding.

ENDS

37

• The comma category of the cospan {∗} → Sets ←− C, where {∗} →Sets chooses the terminal object of Sets;• The opposite of the comma category (よ ↓ dW e), where dW e : {∗} →[C, Sets] is the name of the functor W , i.e. the unique functor choosing
the presheaf W ∈ [C, Sets]:W´(C W )op∗

Cop[C, Sets]

Proof. It is an exercise in Yoneda lemma and universal properties.



Remark 4.4 : There is a fourth characterization for the category of elementsof a presheaf which couches it as a particular weighted colimit which, using
the formalism exposed in this section, can be written as a coend. Example
4.22 below is a guided exercise in proving and clarifying this statement.
´
Proposition 4.5 : The category of elements C W of a functor W : C → Sets
comes equipped with a canonical“Grothendieck fibration” to the domain´
of W , which we denote Σ : C W → C, defined forgetting the distinguishedelement u ∈ W c.
´
Proof. We only have to prove that Σ : C W → C is an isofibration; given an
isomorphism ϕ : x → c = Σ(c, u), we can define v ∈ W x to be W (ϕ−1 )(u).

Notation 4.6 : All along the present section, we assume that V denotes a
Bénabou cosmos, i.e. a symmetric monoidal closed, complete and cocomplete
category which is the “base” for our enriched category theory.
Remark 4.7 : Let F : C → A be a functor between small ordinary categories.
The limit lim F of F can be characterized as the representing object of a←−suitable presheaf: indeed, we have the natural isomorphism
A(a, lim F ) ∼(68)= SetsC (∗, A(a, F ( )))←−where ∗ is a shorthand to denote the terminal functor C → Sets : X 7→ ∗sending every object to the terminal set, and A(a, F ( )) = A(a, F ) is
the functor C → Sets sending c to A(a, F c) (so we represent the functor
a 7→ A(a, F )).
Dually, the colimit lim F can be characterized, in the same notation, as−→
the representing object in the natural isomorphism∼ SetsC (∗, A(F ( ), a)).
A(lim F, a) =
(69)
−→

38

F. LOREGIAN

SetsC (∗, A(F, a)) is a set of natural transformations and a 7→ SetsC (∗, A(F, a))
is a functor: the leading idea behind the definition of weighted co/limit is togeneralize this construction to admit shapes other than the terminal presheaf
for the domain functor. We can package this intuition in the following
definition:
Definition 4.8 [Weighted limit and colimit]: Given functors F : C → A
and W : C → Sets, we define the weighted limit of F by W as a representative
for the functor sending a ∈ A to SetsC (W, A(a, F ( ))), in other words theweighted limit of F by W is an object limW F ∈ A such that←−
A a, limW F ∼= SetsC (W, A(a, F ( )))
←−

(70)

naturally in a ∈ A. Dually we define the colimit of F : C → A weighted by
W : Cop → Sets to be an object limW F ∈ A such that−→

A limW F, a ∼= SetsC (W, A(F ( ), a)).
−→
op

(71)

Example 4.9 : Let f : ∆1 → A the functor choosing an arrow f : x → y inA, and W : ∆1 → Sets the functor sending {0 < 1} to the single arrow{0, 1} → {0}; then a natural transformation W ⇒ A(a, f ) consists ofarrows W 0 → A(a, x), W 1 → A(a0 , y), namely on the choice of two arrows
h, k : a → x such that f h = f k: the universal property for limW f implies←−
that this is the kernel pair of the arrow f , namely that h, k fill in the pullback
a

(72)

h

#

k

limW f
←−
& 

x


/x
y
f



f

/y

Proposition 4.10 [Weighted co/limits as co/ends]: When the indicateduniversal objects (the end below and the Sets-cotensor (X, a) 7→ aX used todefine it) exist, we can express the weighted limit limW F as an end:←−ˆ
SetsC (W, A(m, F )) ∼Sets(W c, A(m, F c))
=
c∈C
ˆ

∼
A m, F cW c
=
c∈C
 ˆ

∼
F cW c= A m,
c∈C

ENDS

39

This string of natural isomorphisms implies that there is a canonical isomorphismˆW∼
lim F =F cW c .
(73)
←−
c∈C

Example 4.11 : Consider the particular case of two parallel functors W, F : C →Sets; then we can easily see that limW F coincides with the set of natural←−transformations W ⇒ F , since the cotensor F cW c amounts to the set
Sets(W c, F c).

Example 4.12 : The ninja Yoneda lemma, rewritten in this notation, saysC( ,c)
that limC(c, ) F ∼F ∼= F c (or, in case F is contravariant, lim= F c).
←−←−
This can be memorized as “representably-weighted co/limits are evaluation”(see Remark 2.6 on the Dirac delta) and suggests that Kan extensions can be
expressed as suitable weighted co/limits, and more precisely that they can becharacterized as those weighted co/limits where the weight is a representablefunctor:ˆ
D( ,K )∼
F.
(74)
RanK F ( ) =F cD( ,Kc) ∼
= lim
←−
c∈C
The following Remark and Proposition constitute a central observation.

Remark 4.13 [ , The Grothendieck construction trivializes weights]:
Definition 4.8 can be extended in the case F : C → A is a V-enriched functor
between V-categories, and W : C → V is a V-co/presheaf; this is the setting
where the notion of a weighted co/limit acquires a supremacy over the“classical” one (where the weight is the terminal presheaf). When V = Sets,
indeed, the Grothendieck construction sending a (co)presheaf into its category of elements turns out to trivialize almost completely the theory of
Sets-weighted limits: as the following discussion shows, in such a situation
every weighted limit can be expressed as a classical (we call them conical,
due to the shape of the weight) limit.
Proposition 4.14 [Sets-weighted
limits are limits]: As shownin Prop.
´´
4.5, the category C W comes equipped with a fibration Σ : C W → C, suchthat for any functor F : C → A one has
limW F ∼lim F ◦ Σ.
(75)
=
←−
←−´
(c,x)∈C W

Proof.
The proof goes by inspection, using the characterization of the end´
W c as an equalizer (see Proposition 1.15), and the characterizationFcc∈Cof Sets-cotensors as iterated products, showing thatˆYY
F cW c ∼F cW c ⇒F c0W c
= eq
c∈C

c∈C

c→c0

40

F. LOREGIAN

∼
= eq
(?) ∼
= eq
∼
=

Y Y


Fc ⇒
Fc ⇒

´
(c,x)∈C Wlim←−´(c,x)∈C W

F c0

c→c0 x∈W cc∈C x∈W c

Y

Y Y

F ◦Σ

Y



´
(c,x)→(c0 ,x0 )∈C W

F c0



(equation (?) is motivated by the fact that every arrow ϕ : Σ(c, x) → c0 has
a unique lift (c, x) → (c0 , x0 ) since W (ϕ)(x) = x0 m).


Remark 4.15 : When we consider Kan extensions as weighted co/limits,
this result agrees with the classical theory: if the weight has the form
W = D(d, K−) for an ´object d ∈ D, and a functor K : C → D, then the
category of elements C W is precisely the comma category (d ↓ K): theright Kan extension of F along K can be computed as the conical limit of
the functor F U , where U : (d ↓ K) → C is the obvious forgetful functor.
Obviously, when every weighted limit exists in A, we can prove that thecorrespondence (W, F ) 7→ limW F is a bifunctor:←−
op× AC −→ A.
(76)lim( ) (=) : SetsC←−
A number of useful corollaries of this fact:• The unique, terminal natural transformation W → ∗ induces acomparison arrow between the weighted limit of any F : C → A
and the classical (conical) limit: lim F → limW F . For example, the←−←−classical limit of the functor f : ∆[1] → A described in Example
4.9 consists of the object a = src(f ); hence the comparison arrowconsists of the unique factorization of two copies of ida along thekernel pair of f .
• The functor lim( ) F is continuous, namely we can prove the sugges←−tive isomorphism→J j F ∼lim −limWj F ,
(77)
= lim
←−
←−J ←−
valid for any small diagram of weights J → [C, Sets] : j 7→ Wj .
lim W

Example 4.16 : Ends can be computed as weighted limits: given H : Cop ×
C → D we can take the hom functor C( , ) : Cop × C → Sets as a weight,
and if the weighted limit exists, we have the chain of isomorphismsˆˆ ˆ 2.1 ˆ0 C(c,c0 ) ∼0 C(c,c0 ) ∼C( , )∼
H(c, c )H(c, c )lim
H=== H(c, c).
←−(c,c0 )∈Cop ×C
c
c0
c
(78)

ENDS

41

Remark 4.17 : This characterization will turn out to be useful during our
discussion of simplicially coherent co/ends. See 7.38.
Remark 4.18 : Weighted colimits are discussed in Exercise E3 and stem froma straightforward dualization process; we refer freely to both concepts fromnow on.
Remark 4.19 : Aside from showing that “weighted limits are the true enrichedcategorical limits”, the above examples and the last characterization of ends
as weighted limits are fundamental steps towards a sensible definition ofenriched ends: given a Bénabou cosmos
V (see Notation 4.6) and a V´functor H : Cop × C → V, we define c H(c, c) to be the limit of H weighted
by C( , =) : Cop  C → V (see Definition 4.25 for the notation C  D).
Remark 4.20 : Let F : C → A, W : C → V, U : Cop → V be V-functors andlet A be V-tensored. There are canonical isomorphisms

W
A(limW F, x) ∼A(F, x)
(79)
= lim
−→
←−
U
W
A(y, lim F ) ∼A(y, F ).
(80)
= lim
←−
←−
This can be recorded in the motto “the hom functor commutes with weighted
limits” (prove it as an exercise).

Example 4.21 [The cone construction as a weighted colimit]: Let
K be a ring, and V = Ch(K) the category of chain complexes of K-modules.
Considering V as a V-category in the obvious way, we aim to prove that the
mapping cone C(f ) = X∗ [1]⊕Y∗ of a chain map f : X∗ → Y∗ [Wei94, 1.5.1] inV can be characterized as the weighted colimit limW f , where f : ∆[1] → V−→
is the arrow f , and W : ∆[1]op → V is the functor which chooses the map
S 1 (K)∗ → D2 (K)∗ , where S n (K)∗ = K[n]∗ is the chain complex with the
only term K concentrated in degree −n, and Dn (K)∗ is the complex
···

/0

···

/0

/0

/K

/0

/ ···

···

/0


/K

K

/0

/ ··· .

ˆ i∈∆1

W (i) ⊗ f (i).

/K

K

/0

/ ··· ,

where the first nonzero term is in degree −n. There is an obvious inclusionS∗n ,→ D∗n+1 :We now aim to prove thatC(f ) ∼
=

(81)

42

F. LOREGIAN

In view of (the dual of) Exercise 1.E10, it is enough to show that there is apushout square/ W (1) ⊗ f (1)W (1) ⊗ f (0)p/ C(f )

W (0) ⊗ f (0)This is a rather simple exercise in universality, given the maps
( 01 )B

/ C(f ) o

0 1
f 0



(82)

A ⊕ A[1].

Example 4.22 [The category of elements of a presheaf]: The scopeof this example is to prove that the category of elements of a functor
F : C → Sets introduced in Def. 4.1 can be characterized as a Cat-weighted
colimit: it is, in particular, the category isomorphic to the colimit
ˆ c∈C
´
C W ∼
c/C × W c(83)=

where W c is the set, regarded as a discrete category; it is, in other words,
isomorphic to the weighted colimit limJ W , where J : Cop → Cat is the−→functor c 7→ c/C (the “coslice” category of arrows
c → x).
´
To prove this statement, we verify that C W has the universal propertyof the coequalizer of the pairaf : a→bb/C × W a

/ a

α
β

/

c∈C

(84)

c/C × W cwhere α has components αf : b/C×W a −−−→ b/C×W b sending
1×F f

h b i↓xf ∗ ×W ah b i↓x, u 7→, F (f )u and β has components βf : b/C × W a −−−−−→ a/C × W asendingh b i↓x, u 7→



f

a−
→b
↓
x





,u .

It’s rather easy to define a functora´a/C × W a −→ C W(85)

ζ(a, u ∈ F a) = ζ(ida , u).

(86)

θ:

a∈C

h a i´having components θa : a/C × W a → C W sending f ↓ , u ∈ F a 7→b(b, F (f )(u) ∈ F b), which coequalizes α and β. This functor
θ has the uni`versal property of the coequalizer:
given
any
other
ζ
:
a/C
× Wa → Ka∈C´
we can define a functor ζ : C W → K such that

ENDS

43

Now notice that every map ζ 0 that coequalizes (α, β) has the property thatζ0

h b ig↓x, F (f )u = ζ 0



f

a−
→b
↓
x





,u

(87)

It is now a routine verification to see that ζ ◦ θa = ζa , and every other functor
with this property must coincide with our ζ. This concludes the proof.
Remark 4.23 : The careful reader may have noticed that all the´ above
discussion gives a fifth presentation for the category of elements C W , as
the image of W under the Kan extension Lanよ J: in the language of §3,
J : Cop → Cat is the nr-context of the paradigm´/: [C, Sets] o
Cat : NJ
(88)
C

where NJ : Cat → [C, Sets] is the “nerve” functor sending A to c 7→ Cat(c/C, A).
´
Remark 4.24 : An´ alternative approach to characterize C W is the following:
the category C W is precisely the lax limit of W regarded as a Cat-valued
presheaf [Kel89, §4], [Gra, Str76].
A fundamental step to write the theory of weighted limits relies upon theabove-mentioned isomorphismA m, limW F ∼(89)= VC (W, A(m, F ( )))←−valid in a V-category A naturally in any object m ∈ A; this isomorphism
has to be interpreted in the base-cosmos V, and this means that we haveto find a way to interpret the category VC as an object [C, V] of V: to do
this, we must endow V Cat with a closed symmetric monoidal structure,such that
V-Fun(C  E, D) ∼(90)
= V-Fun(E, [C, D]).
Definition 4.25 : Given two V-categories C, D we define the V-categoryC  D having• as objects the set C × D, and• as V-object of arrows (c, d) → (c0 , d0 ) the object
C(c, c0 ) ⊗ D(d, d0 ) ∈ V.

(91)

The free V-category I associated to the terminal category is the unit object
for this monoidal structure.
Proposition 4.26 : ( V-Cat, ) is a closed monoidal structure, with internal
hom denoted [ , ] : V-Catop × V-Cat → V-Cat.

Proof. Given C, D ∈ V-Cat we define a V-category whose objects are Vfunctors F, G : C → D and where (‘abstracting’ Theorem 1.19 to the enriched44

F. LOREGIAN

setting) the V-object of natural transformations F ⇒ G is defined via theendˆ[C, D](F, G) :=D(F c, Gc).

(92)

c∈C

Recall that in the unenriched case, the end was better understood as theequalizer of a pair of arrows:
ˆ
Y

Y Y
D(F c, Gc) ∼D(F c, Gc) ⇒D(F c, Gc0 )
(93)
= eq
c∈C

c∈C

c,c0 c→c0

In the enriched
case, we can consider the same symbol, and re-interpret theQ
product C(c,c0 ) as a suitable power in V:
ˆ
Y

Y
0
D(F c, Gc) ∼D(F c, Gc) ⇒D(F c, Gc0 )C(c,c )
(94)
= eq
c∈C

c∈C

c,c0

(see also [Gra80, §2.3], [Dub70] for a more detailed discussion about co/ends
in enriched setting.)
It remains to prove, now, that the isomorphism (90) holds: this is rather
easy, since in the above notations, any functor F : C  E → D defines aunique functor F̂ : E → [C, D].10


The given definition for the enriched end allows us to state an elegant
form of the V-enriched Yoneda lemma:
Remark 4.27 [ V-Yoneda lemma]: Let D be a small V-category, d ∈ D an
object, and F : D → V a V-functor. Then the canonical map
F d −→ [D, V](D(d, ), F )(95)

induced by the universal property of the involved end11 is a V-isomorphism.
Homotopy co/limits as weighted co/limits. We can express theBousfield-Kan construction for the homotopy co/limit functor using co/end
calculus (see 7.2.1 for a crash course on what’s an homotopy co/limit). We
condense Bousfield-Kan construction in the following series of examples.
Theorem 4.28 [The Bousfield-Kan formula for homotopy co/limits]:
Let F : J → M be a diagram in a model category (M, wk, cof, fib) which is
tensored and cotensored over the category sSet of simplicial sets by functors
10Notice that for any two objects e, e0 ∈ E, the collection of arrows hom(e, e0 ) →hom(F (c, e), F (c, e0 )] is a wedge in c ∈ C.
11Notice that this is an alternative point of view on the proof of the ninja Yoneda
lemma 2.1; this arrow is induced by a wedge {F d → V(D(d, a), F a)}a∈C , whose members
are the mates of the various D(d, a) → V(F d, F a) giving the action of F on arrows.

ENDS

45

t : sSetop × M → M and · : sSet × M → M. Then the homotopy limit
holim←−−− F of F can be computed as the endˆ
N (J/j) t F (j),(96)
jand the homotopy colimit −holim−−→ F of F can be computed as the coendˆ j
N (j/J) · F (j).
(97)
Remark 4.29 : These two universal objects are weighted co/limits in an
evident way: it is possible to rewrite them as limN (J/ ) F and limN ( /J) F .
←−−→The idea behind this characterization is that the co/limit functor results
as the weighted colimit over the terminal weight. When we want to pass tothe homotopy invariant version of the lim( ) ( ) bifunctor we can “derive”−→
the diagram part as well as the weight part. Bousfield-Kan formula arises
precisely when we derive the weight: N (j/J) and N (J/j) are contractible
categories, and they are linked to N (∗) by an homotopy equivalence inducedby the terminal functor.
Then, these categories must be thought as proper replacements for thehomotopy co/limit functor.
Example 4.30 : The mapping ` cylinder of f : A → B is the topological space
B from the smallest quotient that identifies
obtained from A × [0, 1](f (a), (a, 0)) for all a ∈ A (compare this example with Example 4.21 andExercise E4).
Exercises for §4´
E1 Prove Equation (77) using the characterization of limW F ∼= c F cW c , plus its←−
universal property.
E2 What is the category of elements of the hom functor? Compare with Definition
1.12.
E3 Every definition we gave until now can be dualized to obtain a theory of weighted
colimits: fill in the details.
(1) (weighted colimits as coends) If A is cocomplete, we can express theweighted colimit limW F as a coend: more precisely−→
ˆ c∈ClimW F ∼Wc · Fc(98)=−→
where we used, like everywhere else, the Sets-tensoring of A.
(2) (left Kan extensions as weighted colimits) Let F : C → A and K : C → D
be functors; thenˆ c∈C
D(K , )
LanK F ( ) ∼D(Kc, ) · F c ∼
F
(99)
=
= lim
−→

46

F. LOREGIAN(3) (coends as hom-weighted colimits) The coend of H : Cop × C → D can be
written as limC( , ) H.
−→
(4) If the weight W is Sets-valued, the ´colimit of F weighted by W can be
written as a conical colimit over Cop W :∼limW F =
lim
FΣ
(100)
−→
−→op ´(c,x)∈CW(5) (functoriality) If the W -colimit of F : C → A always exists, then thecorrespondence (W, F ) 7→ limW F is a functor, cocontinuous in its first−→variable:
lim(
−→

)

op

(=) : SetsC

× AC −→ A,lim Wj→Jlim −F ∼limWj F
(101)
= lim
−→
−→J −→(6) (comparison) There is a canonical natural transformation W → ∗, inducing
a canonical comparison arrow from the W -colimit of any F : C → A to theconical colimit.
E4 Let w : S 0 ,→ D1 be the canonical inclusion of {0, 1} into [0, 1] ⊂ R, with theusual topology; prove that the mapping cone of a continuous map f : X → Y is
precisely the weighted colimit limw f .
−→E5 Fill in the details of the above proof; for those who need a hint, show that
E(e, e0 ) → D(F (x, e), F (x, e0 )) is a wedge in x ∈ C.
LanJ W
E6 Show that there are canonical isomorphisms limW F J ∼F , and dually
= lim
←
−
←
−
WJ
W
∼
lim F = lim LanJ F .
−→−→5. The theory of relators.
The lucid presentation in the notes [Béner] and in [CP08, §4], [Bor94b]
are standard references to follow this section. Our only merit here is having
expressed several proofs using coend calculus. The arguments are essentiallyunchanged, and yet the employment of coends is implicit in [Béner], andonly partially spelled out in [Bor94b].The lucid presentation in the notes[Béner] and in the book [CP08, §4] are standard references to follow thissection. Our only merit here is that we have restated several proofs using
coends; these arguments are somewhat implicit in [Béner] and only partiallyspelled out in [Bor94b].
Definition 5.1 [The bicategory of relators]: There exists a bicategory
Relt having• 0-cells (objects) those of Cat (small categories A, B, C, D, . . . );B, the functors Aop × B →• 1-cells p, q . . . , depicted as arrows A
Sets;• 2-cells α : p ⇒ q the natural transformations between these functors.

ENDS

Given two contiguous 1-cells A
as the coendpq  p(a, c) :=

B

q

ˆ x∈B

47

C we define their composition q  p(102)p(a, x) × q(x, c)Definition 5.2 : This definition works well also with Sets replaced by anarbitrary Bénabou cosmos V, i.e. in any symmetric monoidal closed andbicomplete category: in this case we speak of V-relators in the bicategory
Relt( V).
Remark 5.3 [Naming a category]: The 1-cells of Relt are more often
called profunctors or distributors ([Béner] follows the equation functions :functors = distributions : distributors), correspondences (consider the casewhen V = {0, 1} i.e. where A, B are sets regarded as discrete categories), or
bimodules (consider the case where V = Ab and A, B are rings; see [Gen15]
and several examples below).
As we will see during the present section, the bicategory Relt carries afairly rich structure: it is then quite difficult to choose a name for its 1-cells
able to convey this richness or the main features thereof.
There are several reasons why most of the above choices are unsatisfying:naming the 1-cells of Relt “profunctors” may generate confusion as the prefix
pro-C denotes the pro-completion of a category C, i.e. the collection of all“formal” cofiltered limits of objects of C; then a pro-functor, etymologically,should be an object of the pro-completion of some functor category, which
is not true; justifying the name “distributor” would request a more strict
analogy between p : A → B and distributions in mathematical analysis;correspondence is a name so inflated that it doesn’t conveys any intuition at
all;. . .
While taking note of this situation, we decide to add a new term to thisvast zoology, and we call the 1-cells of Relt “relators”, motivated by theanalogy in the following example.
Example 5.4 [Relators as generalized relations]: A relator between{0, 1}-categories is a function between sets Aop × B → {0, 1}, namely afunction A × B → {0, 1}, or in other words a relation regarded as a subset
R ⊆ A × B.
From this point of view, relators A
B can be thought as generalized
relations, taking values in more complicated, or structured, enriching cosmoi.
This point of view is what Lawvere [Law73, §4,5] calls generalized logic, andit regards the coend in Def. 5.1, and the product p(a, x) × q(x, b) therein,
as a generalized existential quantification and a generalized conjunction
respectively, giving a composition rule for generalized relations: the coend
stands as(x, z) ∈ R ◦ S ⇐⇒ ∃y ∈ Y : (x, y) ∈ R ∧ (y, z) ∈ S



(103)

48

F. LOREGIAN

for two relations R ⊆ X × Y, S ⊆ Y × Z: we can depict this analogy as in
Figure 5: this is valid for suitable relations R : X
Y and S : Y
Z orsuitable relators r : X
Y and s : Y → Z.

(x, z) ∈ S ◦ R⇐⇒∃y ∈ Y(x, y) ∈ R(s  r)(x, z)

=

ˆ y∈Y

r(x, y)∧(y, z) ∈ S

×s(y, z)

Figure 2. The analogy between the composition of relators
between categories and the composition of relations between
sets gives rise to what lawvere calls generalized logic.

Example 5.5 : Let A, B be sets, considered as categories having only identity
arrows via the embedding Sets ⊂ Cat. A relator A
B is then simply acollection of sets Pab , one for each a ∈ A, b ∈ B. Their composition thenresults in a “categorified” matrix multiplication, in that the coend simplifies
to be a mere coproduct: given p : A → B, q : B → C we have(p  q)ac =if Pab = p(a, b) and Qbc = q(b, c).

ab∈BPab × Qbc

(104)

Remark 5.6 : There is an alternative, but equivalent definition for q  p
b as a free cocompletion: any
which exploits the universal property of C
relator p : A
B can be identified with its mate under the adjunction giving
the cartesian closed structure of Cat,
Fun(Aop × B, Sets) ∼= Fun(B, [Aop , Sets])(105)

b obtained as b 7→ p( , b). Hence we can defineb: B → Ai.e. with a functor p
q
p
b◦q
b:
the composition A
B
C to be Lanよ p

B

よ

C

b
q


b
/B

b
p / b
@A

Lanよ b
p

(106)

ENDS

49

This is equivalent to the previous definition, in view of the characterizationb given in Equation 39:
of a left Kan extension as a coend in A,
ˆ b
b
∼
b
b(b).
Lanよ p =B(よ(107)b, ) · p
Since in Sets copower coincides with product (i.e. X · Y ∼= X × Y , since∼∼Sets(X · Y, B) = Sets(X, Sets(Y, B)) = Sets(X × Y, B), naturally in B), we
have
ˆ b
b
b (q
b (c)) ∼b (c)) · pb(b)Lanよ p
B(よ
=
b, q
ˆ b
∼
b (c)(b) · p
b(b)
q
=
ˆ b
∼
p( , b) × q(b, c).
=Remark 5.7 : The properties of (strong) associativity and unitality for thecomposition of relators follow directly from the associativity of cartesian
product, its cocontinuity as a functor of a fixed variable, and from the ninja
Yoneda lemma 2.1, as shown by the following computation:• Composition of relators is associative (up to isomorphism), giving
the associator of a bicategory structure:ˆ xp  (q  h) =p(b, x) × (q  h)(x, a)
ˆ x
ˆ y

=
p(b, x) ×q(x, y) × h(y, a)
ˆ xy


∼
p(b, x) × q(x, y) × h(y, a)=ˆ x(p  q)  h =(p  q)(b, x) × h(x, a)ˆ xy 
∼p(b, y) × q(y, x) × h(x, a)=and these results are clearly isomorphic, and naturally so, once
we changed name to “integration” variables (which are obviously“mute”). See Remark 5.8 below for a discussion on the coherence
laws of this associator.
• Any object A has an identity arrow, given by the “diagonal” relatorA( , ) = homA : Aop × A → Sets: the fact that p  hom ∼= p,
hom  q ∼= q simply rewrites the ninja Yoneda lemma.

Remark 5.8 : The isomorphism above is part of the data turning Relt intoa bicategory; the associator realizes the identification between different

50

F. LOREGIAN

parenthesizations of 1-cells, and the unitor realizes the identification betweenp  hom ∼= p.
To ensure that “every” diagram which can be constructed from these data
commutes some coherence conditions have to be imposed. One of these is
the pentagon identity [ML98], encoded in the following diagram:(wx)(yz)((wx)y)zw(x(yz))(w(xy))zw((xy)z)

This equality is natural in each argument (as a consequence of being acomposition of natural transformations).
It’s immediate to observe that the validity of the pentagon identity in thecase of the cartesian monoidal structure of Sets, and the naturality thereof,
ensure that the associator (whose components are) (p  q)  h ⇒ p  (q  h)satisfies the pentagon identity; a similar argument shows that the unitor
satisfies similar (left and right) triangular identities, as a consequence of thenaturality of the ninja Yoneda lemma 2.1.
Definition 5.9 [Einstein notation]: There is a useful notation which canbe implied to shorten involved computations with coends, and which isparticularly evocative when dealing with relators; we choose to call it Einsteinconvention for evident graphical reasons.12
Let p : A
B, q : B
C be two composable relators. If we adopt thenotation pab , qbc to denote the images p(a, b), q(b, c) ∈ Sets (keeping trackthat superscripts are contravariant and subscripts are covariant components),
then composition of relators acquires again the form of a “matrix product”:
p  q(a, c) =´bpab × qbc =´bpab qbc .

(108)

From now on, we feel free to adopt the Einstein summation convention
during long calculations.
12This notation has been adopted also in the beautiful [RV14], a valuable reading forseveral reasons, last but not least the fact that its authors adopt some coend-fu to simplify
the discussion about “Reedy calculus”.

ENDS

51

Remark 5.10 [(Co)presheaves are relators]: Presheaves on C obviously correspond to relators C
1; copresheaves, i.e. functors C → Sets,
correspond to relators 1
C.
5.1. Embeddings and adjoints. There are two identity-on-objects embeddings Cat → Relt (respectively the covariant and the contravariant one,looking at the behaviour on 2-cells), and send a diagram in Cat respectively
to
C
α

F



D

DCZ

Catop → Relt

+3 G


7→

pF

pα

C
+3

pG

α

F



D

D

C

Catco → Relt

+3 G


7→

sk p

pF



α

D

pG



(109)
This clearly defines a (pseudo)functor, since it’s easy to see that
• pF G ∼= pF  pG , and pF G ∼= pG  pF ;id
A• pidA = p = A( , ).
Natural transformations α : F ⇒ G are obviously sent to 2-cells in Relt, andthe covariancy of this assignment is uniquely determined as in the diagram
above.
Remark 5.11 : The 1-cells pF , pF are not independent: they are adjoint
1-cells in the bicategory Relt. Indeed, for every F ∈ Cat(A, B) we can define
2-cells = F : pF  pF

η = ηF : A( , )+3 B( ,)

(110)

+3 pF  pF

(111)

(counit and unit of the adjunction): here we unravel the coends involved inthese definitions.
• For what concerns the counit, we write the coend pF  pF as thequotient setˆ xaB(a, F x) × B(F x, b) / '(112)
B(a, F x) × B(F x, b) =x∈Awhere ' is the equivalence relation generated by a −→ F x, F x −
→
u



u0

v0



v

b ' a −→ F y, F y −→ b if there is t : x → y such that v 0 = F t ◦ v
0e F t ◦ u = u . This can be visualized as the commutativity of the

52

F. LOREGIAN

square
u

8 Fx

a

&

Ft
u0

%



Fy

(113)

v

9b

v0



Now it’s easily seen that sending a −→ F x, F x −→ b in the composition v ◦ u descend to the quotient with respect to ', hence : pF  pF → B( , ) is well defined. All boils down to notice thatthe compositionuv

c : B(a, F x) × B(F x, b) → B(a, b)(114)

defines a cowedge in the variable x.
• The unit of the adjunction is the 2-cellη : A( , )(115)+3 pF  pF

´X
obtained when we noticed that pF  pF (a, b) =B(F a, x) ×
B(x, F b) ∼= B(F a, F b) (as a consequence of the ninja Yoneda lemma),
is simply determined by the action of F on arrows, A(a, b) →B(F a, F b).
We now have to verify that the zig-zag identities (see [Bor94a, Thm.3.1.5.(2)])
hold:(pF  ) ◦ (η  pF ) = idpF

(  pF ) ◦ (pF  η) = idpF
As for the first, we must verify that the diagram
pF
pF o

∼

∼

/ B( ,)  pFpF  B( , ) oηpF/ (pF  pF )  pF


pF 

(116)

∼
=

pF  (pF  pF )commutes.
One has to send h ∈ pF (u, v) = B(F u, v) in the class [(idu , h)] ∈´xF(idu ), h)] ∈´ xyB(u, x) × B(F x, v), which must go under η  p in the class´[(FyB(F a, x)×B(x, F y)×B(F y, b), canonically identified with
B(F a, F y)×
B(F y, b). Now pF   acts composing the two arrows, and one obtains
F (idA ) ◦ h = h back.

ENDS

53

Similarly, to prove the second identity, the diagram
pF
pF o

∼

∼

/ pF  B( ,)B( , )  pF o

pF η

/ pF  (pF  pF )


pF

(117)

∼
=

(pF  pF )  pF

must commute (all the unlabeled isomorphisms are the canonical ones). Thistranslates into




u
a−
→ Fb / (u, idb )∼ 

/ (u, F (idb )) 

/ u ◦ F (idb ) = u,which is what we want; hence pF a pF .

(118)


Remark 5.12 : Two functors F : A  B : G are adjoints if and only if pF ∼= pG(and therewith G a F ) or pG ∼= pF (and therewith F a G).

Remark 5.13 : It is a well-known fact (see [Bor94a, dual of Prop. 3.4.1]) that
if F a G, then F is fully faithful if and only if the unit of the adjunction
η : 1 → GF is an isomorphism.
This criterion can be extended also to functors which do not admita “real” right adjoint, once noticed that F is fully faithful if and only
if A(a, b) ∼= B(F a, F b) for any two a, b ∈ A, i.e. if and only if the unit
η : homA ⇒ pF  pF is an isomorphism.
Example 5.14 : Given a relator p : A
B and a functor F : B → D we canb (provided thisdefine p ⊗ F to be the functor A → D given by Lany F ◦ p
b is the adjunct of p.
b: B → A
colimit exists), where p
More explicitly,
ˆ b
ˆ b
∼
p ⊗ F (a) =Nat(yb , p( , a)) · F b =pba · Fb
(119)

Exploiting this definition, several things can be proved via coend-fu:• homB ⊗F ∼= F as a consequence of the ninja Yoneda lemma;qpF• If CA
B−→ X, then (p  q) ⊗ F ∼= q ⊗ (p ⊗ F ): indeedˆ b[(p  q) ⊗ F ]a =(p  q)ba × Fb
ˆ bx∼
pbx × qxa × Fb
=
ˆ x
ˆ b

x
∼
qa ×
pbx × Fb=ˆ x∼
qxa × (F ⊗ p)x = [q ⊗ (p ⊗ F )]a=

54

F. LOREGIAN

Example 5.15 [Kan extensions in Relt]: Any relator p has a right Kan
extension Ranp in the sense that the notion has in any bicategory, wherecomposition of functors or natural transformations is replaced by compositionof 1- or 2-cells.
One has the following chain of isomorphisms in Relt (see Definition 5.9
for the Einstein convention):ˆ∼Nat(g  p, h) =Sets (g  p)ab , hab
ˆab
ˆ x

∼
Sets
gax × pxb , hab
=
ˆab

∼
Sets gax , Sets(pxb , hab )
=
ˆabx
ˆ


a
∼
Sets gx , Sets(pxb , hab )
=
b
ˆax


∼
Sets gax , Ranp hax=ax∼= Nat(g, Ranp h)when we define Ranp h(a, x) to be Nat p(x, ), h(a, ) .
Remark 5.16 [The multibicategory of relators]: The bicategory of
relators can be promoted to a multibicategory in the sense of [CKS03, 1.4];
this means that we exploit the (partial) monoidal structure on each Relt(C, D)
to specify a class of multimorphisms η : p1 , . . . , pnq, depicted as diagrams
X0

p1

p2

/ X1

/ ...

pn

/ Xn
C

(120)

η




q

and composition, associativity and unitality thereof, follow at once frompasting laws for 2-cells in 2-categories [Kel82] (try to outline them as astraightforward exercise).
Exercises for §5
E1 Describe relators between monoids, regarded as one-object categories; describe
relators between posets regarded as thin categories.
E2 Given relators k : C
D and l : C
E defineˆk B l = [k(c, ), l(c, )]
c

ENDS

55

Show that this operation is a Kan lifting (of l along k); dually, given h : DAand l : E
A we can defineˆ(121)
l C h = [h(=, a), l( , a)].
aShow that this second operation is a Kan extension (some vagueness is intended
to be fixed as part of the exercise), and that these two operations “behave likean action” on (Relt, , hom) on the bicategory Relt:i) (k  h) B l ∼= k B (h B l);ii) l C (k  h) ∼= (l C k) C h;iii) hom Bl ∼=l∼
= l C hom.
This is a naïve way to see that the structure on Relt given by  is biclosed (i.e., is a bifunctor Relt(A, B) × Relt(B, C) → Relt(A, C) and each p  , as well as
each  q have right adjoints).
E3 The collage of two categories A, B along a relator p : A
B is defined to be thecategory A ]p B with the same objects as A q B and morphisms given by the ruleA ]p B(x, y) =A(x, y)if x, y ∈ A
if x, y ∈ B
if x ∈ A, y ∈ B

B(x, y)p(x, y)(122)

and empty in every other case. Show that A ]p B has the universal property of
the category of elements of p, regarded as a presheaf.
E4 Show that the composition laws p(A, B) × B(B, B 0 ) → p(A, B 0 ), A(A, A0 ) ×p(A0 , B) → p(A, B) of arrows in A ]p B are governed by the universal property
of a coend.
FG−A−→ Y is defined to be theE5 The cocomma object (F/G) of two functors X ←pushout of/ A × ∆[1]
AqA
(123)



XqY
in Cat, where the horizontal arrow is the “cylinder” embedding. Show that
(F/G) is the collage of X and Y along the relator pG  pF : X
Y.
pq
E6 Given relators A
B
C consider the categories A ]ϕ B and B ]ψ C. Describe
the pushout/ A ]ϕ B
B
(124)



B ]ψ Cp/H

in Cat. Is there a relation between H and the collage A ] C along q  p?

6. Operads using coends.
Since they were introduced by P. May in his [May72] to solve a problem
in algebraic topology,13 it has been clear that operads are monoid-like objects
13
There are several reasons why algebraic topologists are interested in spaces Y which
are homotopy equivalent to ΩX; they are much more interested in spaces Y ' Ωn X, and56

F. LOREGIAN

in some category of functors; making this analogy a precise statement, using
the power of coend-fu, is the content of Kelly’s [Kel05], which we follow here
almost verbatim.
A certain acquaintance with the machinery of operads is a fundamental
prerequisite to follow the discussion; unfortunately, given the plethora ofdifferent interpretation of the theory, and different areas of mathematics
where the notion of operad arises, the beginners (the author of the present
note is undoubtedly among them) may feel rather disoriented when approaching any book on the subject, so it’s extremely difficult to advise a single,
comprehensive reference.
Among classical textbooks, we can’t help but mention the founder [May72],
as well as more recent monographies like [LV12, MSS02] written respectively
from the algebraist’s and topologist’s point of view. Among less classicaland yet extremely valid points of view, the author profited a lot from a lucid,
and unfortunately still unfinished, online draft [Tria] written by T. Trimble.
Local conventions. Along the whole section we will adopt the following
notation and conventions:• P is the groupoid of natural numbers, i.e. the category having objects
the nonempty sets {1, . . . , n} (denoted as n for short, assuming that0 = ∅) where P(m, n) = ∅ if n 6= m and Sn (the group of bijectionsof n-elementsets) if n = m. It is evident that P is the disjoint union`of groups n≥0 Sn in the category Gpd of groupoids.
• V is a fixed Bénabou cosmos (i.e. a bicomplete closed symmetricmonoidal category, “a good setting to do enriched category theory”,
see Notation 4.6).
Notice that P has a symmetric monoidal structure, with tensor the sum ofnatural numbers; the action on arrows is given by (σ, τ ) 7→ σ + τ defined
acting as σ on the set {1, . . . , m} and as τ on the set {m + 1, . . . , m + n}(these permutations are called shuffles).
6.1. Convolution product. We begin our discussion presenting a general
theorem on monoidal categories, first outlined by B. Day: it will be utterly
generalized in our Appendix A.
The rough idea is the following: in the same way the set of regularfunctions f : G → C ´on a topological group G acquires a convolution productgiven by (f, g)(x) = G f (xy −1 )g(y)dy (the integral sign here is not an end!),we can endow the category of co/presheaves F : C → V with a monoidal
structure induced by a monoidal structure on C, which is different from thepointwise one, induced by the monoidality of the codomain. This is calledin spaces such that “Y ' Ω∞ X”: these are called infinite loop spaces. [May72] offers away to recognize infinite loop spaces among all spaces. See [Ada78] for more informations.

ENDS

57

the convolution product of functors; appendix A will give a generalization ofthis point of view in form of exercises (see in particular Exercises A.4, A.5).
Definition 6.1 [Day convolution]: If C is a symmetric monoidal category,then the functor category [C, V] is itself a Bénabou cosmos with respect to themonoidal structure given by Day convolution product: given F, G ∈ [C, V]
we defineˆF ∗ G :=

cd

C(c ⊗ d, ) · F c ⊗ Gd

(125)

where we recall that X · V for X ∈ Sets, V ∈ V is the copower (or tensor)
X · V such that
V(X · V, W ) ∼(126)= Sets(X, V(V, W )).

Notation 6.2 : In the following sections we will make use of the “Einsteinnotation” for co/ends defined in 5.9; this will compactify a lot the exposition.
A fundamental rule to avoid getting lost is the following, really akin to the
Einstein convention for tensor operations: variables of integration are paired
as subscript-superscript, and whenever they are paired a co/end operation
is implicit.
Without smart ideas to specify the difference we are forced to maintainall integral signs, to discern ends from coends. In Einstein notation we write
the convolution as
ˆ
F ∗G=

cd

Cc⊗d
( ) Fc Gd(127)

Proof. We have to show that this really defines a monoidal structure:• Associativity follows from the associativity of the tensor product on
C and the ninja Yoneda lemma (see the remark above for the Einsteinconvention; it is also harmless to suppress the distinction between
monoidal products in V and Sets-tensors, since the distinction canbe easily devised with a “dimensionality check”):
ˆ ab[F ∗ (G ∗ H)]x =Cxa⊗b Fa (G ∗ H)b
ˆ ab
ˆ cd
a⊗b
∼
Cx
Cc⊗d
=
b Fa Gc Hd
ˆ abcdc⊗d∼Ca⊗b=x Cb Fa Gc Hd
ˆ acd∼
Ca⊗(c⊗d)Fa Gc Hd
=
x
ˆ acd
∼
[(F ∗ G) ∗ H]x =C(a⊗c)⊗dFa Gc Hd .
x

58

F. LOREGIAN• (Right) unitality : choose J = よI = C(I, ) and notice that theninja Yoneda lemma implies that
ˆ cd∼
[F ∗ J]x =Cc⊗dx Fc Jd
ˆ cd
I
∼
Cc⊗d
=
x Cd Fc
ˆ c
∼
∼
Cc⊗I
=
x Fc = Fx .
• Left unitality is totally analogous.

Example 6.3 [Subdivision and joins as convolutions]: Compare Example 3.15 and the definition of join of augmented 14 simplicial sets given in[Joy08]: given X, Y ∈ sSet+ we define
ˆ p,q
(128)
X ?Y =
Xp × Yq × ∆ ( , p ⊕ q)
where ⊕ is the ordinal sum operation (see again [Joy08] or rather our 3.15).
The category [C, V] is left and right closed with respect to this monoidal
structure: the exponential G/H (or rather the functor G/ which is rightadjoint to ∗ G) is given byˆ(129)
G/H := [Gc, H(c ⊗ )]cwhere [ , ] is the internal hom in V, and often denoted JG, HK. We cancompute directly thatˆ∼
Nat(F ∗ G, H) =V (F ∗ G)c, Hc
c

∼
=

ˆ

∼
=

ˆ

∼
=

V

c

ˆabc
abc

ˆ ab

Ca⊗b
c Fa Gb , Hc

V Cca⊗b Fa Gb , Hc



!



V Fa , [Ca⊗bc Gb , Hc ]14The category ∆ lacks an initial object [−1] = ∅; if we add this colimit we geta category ∆ + , and an augmented simplicial set is a presheaf on ∆ + ; the category ofaugmented simplicial sets is denoted sSet+ . There is a triple of adjoints induced by the
inclusion i : ∆ ⊂ ∆ + and linking the categories of simplicial and augmented simplicial sets.
The join operation is easily seen to restrict to simplicial sets (identified with augmented
simplicial sets with empty set of (−1)-simplices) giving a monoidal structure on sSet.

ENDS

ˆ

59





V Fa , Gb , [Ca⊗bc , Hc ]
abc
ˆ
ˆ


∼
V Fa , Gb , [Ca⊗b=c , Hc ]cˆab∼V Fa , [Gb , Ha⊗b ]
=
ˆ
ˆab 

∼
V Fa , [Gb , Ha⊗b ]=b
ˆa∼V(Fa , JG, HKa )
=
∼
=

a


∼
= Nat F, JG, HK .

Remark 6.4 : In the particular case C = P this means that [P, V] is monoidal
closed if we defineˆ mn(F ∗ G)k :=P(m + n, k) · Fm ⊗ Gn
ˆJF, GKk := [Fn , Gn+k ]nIn particular, we have the formulaˆ k1 ,...,kn  X· F1 k1 ⊗ · · · ⊗ Fn kn .
ki ,F1 ∗ · · · ∗ Fn =
P

(130)

for the iterated convolution of F1 , . . . , Fn ∈ [C, V], which will become usefulin a while.
The gist of the definition of a V-operad lies in an additional monoidal
structure on [P, V], defined by means of the Day convolution:
Definition 6.5 [Diamond product on [P, V]]: Let F, G ∈ [P, V]. Defineˆ mF  G :=F m ⊗ G∗m ,(131)
where G∗m := G ∗ · · · ∗ G.

Associativity exploits the following

Lemma 6.6 : There exists a natural equivalence (F  G)∗m ∼= F ∗m  G.

Proof. It’s a formal manipulation:
ˆ ~ni  X
(F  G)∗m =Pni ,· (F  G)n1 ⊗ · · · ⊗ (F  G)nm
ˆ ~ni ,ki  X

∼
P
ni ,
· F k1 ⊗ G∗k1 n1 ⊗ · · · ⊗ F km ⊗ G∗km nm=

60

F. LOREGIAN∼=ˆ ~ni ,~ki

∼
=

ˆ ~ki

F k1 ⊗ · · · ⊗ F km ⊗ G∗k1 ∗ · · · ∗ G∗km

∼
=

ˆ ~ki

F k1 ⊗ · · · ⊗ F km ⊗ G∗

X

F k1 ⊗ · · · ⊗ F km ⊗ P

P

ni ,



· G∗k1 n1 ⊗ . . . G∗km nmkiˆ ~ki ,r  XPki , r ⊗ F k1 ⊗ · · · ⊗ F km ⊗ G∗r
ˆ r∼
F ∗m r ∗ Gr = F ∗m  G=ninja ∼=

´ ~ni ´ n1 ,...,nm(we used a compact notationfor=; the ninja Yoneda Lemma isˆ r∗n ∼used in the form G =P(n, t) · Gt = P(n, )  G, because (n, G) 7→ G∗n
is a bifunctor P × [P, V] → [P, V]).



Associativity of the diamond product now follows at once: we haveˆ m(F  (G  H))(k) =F m ⊗ (G  H)∗m kˆ m∼
F m ⊗ (G∗m  H)k
=
ˆ m,l
∼
F m ⊗ G∗m l ⊗ H ∗l k=
ˆ l∼(F  G)l ⊗ H ∗l k=
= ((F  G)  H)(k).

A unit object for the -product is J = P(1, )·I; indeed J(1) = I, J(n) = ∅ V
for any n 6= 1 and the ninja Yoneda lemma applies on both sides to show
unitality rules:• On the left one has
ˆ m
ˆ m
∗m
J F =Jm ⊗ F=
P(1, m) · F ∗m ∼(132)
= F ∗1 = F.
• On the right, G  J ∼= G once noticed that J ∗m ∼= P(m, ) · I sinceˆ ~ni  X

· P(1, n1 ) · · · · · P(1, nm ) · I
J ∗m =Pni ,∼ P(1 + · · · + 1, ) · I = P(m, ) · I
ninja =

ENDS

61

becauseˆ ~niP(n1 +· · ·+nm , )·P(1, ni ) ∼= P(n1 +· · ·+ni−1 +1+ni+1 +· · ·+nm , ),(133)
for any 1 ≤ i ≤ m (it is again an instance of the ninja Yoneda
Lemma). One has
ˆ m
ˆ m
∗m ∼
(134)
GJ =
Gm ⊗ J =Gm ⊗ P(m, ) · I ∼= G.

Theorem 6.7 : The -monoidal structure is left closed, but not right closed.
Proof. It is a formal manipulation:

ˆ mNat(F  G, H) ∼Nat
F m ⊗ G∗m , H
=
ˆ

ˆ m
∼
V
F m ⊗ G∗m , H=ˆk∼V(F m, [G∗m k, Hk])
=
km
ˆ
ˆ
∼
V(F m, [G∗m k, Hk])=kmwhich is equal to Nat(F, {G, H}) if we define {G, H}m =ˆ[G∗m k, Hk].

k

Hence the functor ( )  G has a right adjoint for any G.
The functor F  ( ) can’t have such an adjoint (Incidentally, this shows
also that the diamond product can’t come from a convolution product withrespect to a promonoidal structure in the sense of Proposition A.3. Re-readthis result after having gone through Appendix A!). We leave the reader
think about the reason.


Definition 6.8 : An operad in V consists of a monoid object in [P, V] endowedwith the (, { , }) left-closed monoidal structure.
More explicitly, an operad is a functor T ∈ [P, V] endowed with a naturaltransformation called multiplication, µ : T  T → T and a unit η : J → T
such thatT T T

T µ

T T
µ

µT

T T

µ

T

are commutative diagrams.

J T

ηT

T T
µ

∼
=

T

T η

∼
=

T J

62

F. LOREGIAN

Definition 6.9 [Endomorphism operad]: For any F ∈ [P, V] the object{F, F } is an operad whose multiplication is the adjunct of the arrow
1ev{F, F }  {F, F }  F

ev{F, F }  F

Fand whose unit is the adjunct of the isomorphism J  F ∼= F.

Unraveling the previous definition, we can notice that an operad in V
consists of• Giving a natural transformation η : J → T amounts to a map η1 : I →T (1), since J(1) = I, J(n) = ∅ for n 6= 1;• Giving a natural transformation µ : T  T → T , in view of theuniversal property of the two coends involved, amounts to give a
cowedgeT m ⊗ P(n1 + · · · + nm , k) · T n1 ⊗ · · · ⊗ T nm

τTkfor any m, n1 , . . . , nm , k ∈ N, natural in k and the ni and such thatthe following diagram commutes:σ∗T m ⊗ P(~n, k) · T ~n

T m ⊗ P(~n, k) · T ~nσ∗

T m ⊗ P(~n, k) · T ~nTk(the notation is self-evident) for every morphism σ ∈ P. This isequivalent to a transformation
T m ⊗ T n1 ⊗ · · · ⊗ T nmb
τ[P(n1 + · · · + nm , ), T ( )](considering the ni fixed and the first functor constant in k) i.e., by
the Yoneda Lemma a natural transformation
T m ⊗ T n1 ⊗ · · · ⊗ T nmbτ

T (n1 + · · · + nm ).

This concludes the discussion, as it is precisely the definition of operad given
in [May97]. It only remains to verify that all the axioms given there aresatisfied. This is a tedious but necessary exercise.

ENDS

63

Exercises for §6
E1 Show that the convolution product on [C, V] results as the following left Kan
extension:
C×C

F ×G

/ V× V

⊗V

/V
<

(135)



E2 Define two functorsCΦ : [P, V] → V(evaluation at 0)Ψ : V → [P, V](the left adjoint to Φ)∼
Prove that Ψ(a ⊗ b) = Ψa ⊗ Ψb and Φ ◦ Ψ ∼= 1; finally, if a ∈ V is identified withthe constant functor in a, then [P, V](a ∗ F, G) ∼= V(a, [F, G]), whereˆ[F, G] :=V(F n, Gn).
(136)n

7. Higher dimensional coend calculus.
Entonces desaparecerán del planeta el inglés y elfrancés y el mero español. El mundo será Tlön. Yo nohago caso, yo sigo revisando en los quietos días delhotel de Adrogué una indecisa traducción quevediana(que no pienso dar a la imprenta) del Urn Burial de
Browne.
J.L. Borges, Tlön, Uqbar, Orbis Tertius

Category theory was born in 1945 when Mac Lane and Eilenberg [EM45]
isolated the correct definition of natural transformation between functors.
By doing this, they introduced the paradigmatic example of a 2-category:in this precise sense then higher category theory is a field as old as category
theory itself. And yet, despite its age, it remains an area where even basic
questions, burdened by an intrinsic computational difficulty, are still intricate,
subtle and very challenging.
Even though for many years higher-dimensional category theory remained
confined to well-defined geographical areas, the last 15 years witnessed asuper-exponential growth of interest across several areas of mathematics:
higher categorical structures have been recognized to lie at the heart ofmodern approaches to geometry [TV05], [Lur09], [BZFN10], logic [Uni13],
topology and mathematical physics [Sch13]; the slow but constant diffusionof a dialect which is powerful enough to encompass all these developments,
and yet sufficiently simple to be studied led to the present situation and led
to a fairly general “reinterpretation” of known theories in a new language,
inspired by homotopy theory. Higher category theory and the fundamental

64

F. LOREGIAN

constructions therein are hence interpreted in a homotopy-invariant way.15
The process of passing from a 1-categorical (also called “classical” in thefollowing) setting to an higher-categorical one can be seen as a process of“heightening”.
Co/end calculus, as a part of the categorical toolbox, makes no exception
and can be heightened (in fact, in several ways).
The scope of the present chapter is to give a compact but lucid presentationof this “higher co/end-fu”. The struggle here is on two separate and opposing
fields: on one side, “ancient” higher category theory [GPS95, Hof11] withits baroque equational approach to coherence conditions is a true nightmare,both for the listener and the exposer. On the other side the “new” approach
to higher category theory based on homotopy theory (mainly that of simplicial sets) reinterprets those very coherence conditions allowing a precious
bookkeeping device, which is nevertheless often too far from the taste ofsome practitioners of categorical algebra.
This simple observation does not distance the author from the current
fashion and the current faith: the “homotopical” approach to higher categories has proved itself a valid tool to actually do beautiful mathematics,
and speaks a subtle and intricate language, forbidden to the inhabitantsof the 1-dimensional world. Nevertheless, we feel this is the right place toclarify our position in the battlefield.
At the moment of writing this note we (as a community) witness several
attempts to acquire a deeper understanding of the landscape of current
mathematics at the level of its fundamental architecture; category theory,
with its indisputable unification power, can be encoded in homotopy theory,and this is part of a philosophical turnaround which takes the notion ofhomotopy as a primitive idea interacting with the similarly primitive notionof structure and operation on a space (seen from this perspective, sets are
discrete groupoids and hence particular cases of topological spaces): there isa growing feeling that any attempt to rewrite a piece of “old” mathematics
turning it into an “homotopically meaningful” statement is, in a suitablesense, a piece of higher category theory.
This is a peculiar, transitional moment in pure mathematics; several
generations used to think in terms of set theory resist the revolution, thinking
that the homotopy groups of spheres are too complicated an object to “lieat the foundation” of current mathematics. This point of view in some sense
15One of the main tenets of higher category theory is that, whatever they are, theseobjects live in the world of (abstract) homotopy theory. There are several ways to justify
this apparently strange remark, but this margin is too narrow to contain any of them: thestarting point for most of them is that the nerve functor Ni (as described in §3) provides
a fully faithful embedding Cat ⊆ sSet; the wild variety of model structures on the categoryof simplicial sets becomes then a tool to better understand category theory.

ENDS

65

indisputable, and yet the author feels that it underlies a subtle epistemologicalposition (what is “simple mathematics”? What has, and what has not, theright to be considered a primitive entity? What is, in the end, the “right”
primitive entity to choose for a foundation of mathematics?) which has nohope to be solved ever (so a fortiori, not in the space of this paragraph).
At the very least, this suggests that the extremely positive attitude towards
the outreach of category theory among the barbarians16 must be taken witha grain of salt. Categorical jargon gets more and more acknowledged as more
and more applications for it are found, and yet it still suffers from a certain
ostracism by some parts of the mathematical community. This is not, or atleast not completely, their fault, and results in a generalized discomfort.
Together with the development of the language, nobody cared about acomplementary development of self-contained presentations, as elementary
as possible, forgetting how this is essential for a rapid and “ healthy ” spreadof ideas. The canonical sources in which by now higher category theory has
been distilled are difficult and time-consuming readings; this leaves many
people behind. The technical subtleties, the intrinsic difficulty of simplicial
homotopy theory and the lack of a meta-language shared by its users makes
the big picture visible only to a handful of people, often the same creatorsof the theory; this is a deterrent for others to come in.
In this age where categorical language is expanding, it is more necessary
than ever to write, to write well, and write simply, addressing to everyone.
It is a deep author’s conviction that by doing this the overall health ofmathematical practice significantly improves, as well as the prompt andfaithful diffusion of profound ideas that would otherwise be exclusively
dedicated to an élite. Rather, it is the consequence of a precise historicalmoment and ideology. This kind of commitment, carried out in the past,
allowed generations of totally unprepared students to appreciate the depthof rich and complex definitions (now for real, undoubtedly, a root of themathematical language essential to every practitioner).
As in all “critical” moments –in which you carefully consider the risks
and benefits of a language-shift– we should be oriented towards ideologicalopenness and a multilingual attitude: this simple pedagogical principle isunfortunately often forgotten by innovators of all ages.
It is difficult to foresee where the current landscape will lead us; eventually,
the world will become Tlön. All we can do then is wait, and have fun learning
something beautiful.

16
The word here only means “the people living outside the polis of category theory”,
without any pejorative meaning.

66

F. LOREGIAN

7.1. 2-coends. One of the most immediate generalization of co/end-fu lives
in the world of 2-categories, where “1-cells” are allowed to be transformedby “2-cells”.
In view of Prop. 1.2, to define 2-co/ends we must exploit some flavour oflimit and colimit in 2-category theory: this calculus has a rather naturalinterpretation in terms of enriched category theory ([KS74] and [Kel89], forexample, offer an invaluably complete survey on this topic, but the reader
should be aware that 2-category theory is not the theory of Cat-enrichedcategories). Here the reader is also warned that the following discussion has
little or no hope to be a self-contained exposition, and instead heavily relies
on classical sources as [Kel82, Dub70].
Notation 7.1 : As always when dealing with higher dimensional cells and theircompositions, there are several “flavours” in which one can weaken strict
commutativity: besides this strictness (where every diagram commutes with
an implicit identity 2-cell filling it), there is a notion of strong commutativity
and universality, where filling 2-cells are requested to be invertible, and lax
commutativity/universality, where 2-cells are possibly non-invertible.17
The definition of lax end of a 2-functor S (given in terms of a universal..
lax wedge ω : b −→ S) is the most general and less symmetric one can give (itcan still be dualized to a colax coend, though!); definitions and theorems
in this subsection are designed and proved in such a way to reduce to thestrong and strict cases as particular examples of lax co/ends, where filling
2-cells are invertible (resp., identities).
Local notation. For the rest of the section we adopt some local conventions:first of all, we denote A 99K B a lax functor between two 2-categories A, B;
as sketched above, this means that we have a correspondence F0 : Ob(A) →Ob(B) and a correspondence F1 : hom1 (A) → hom1 (B) at the level of 1-cells,such that there exist “laxity cells” F (g)F (f ) ⇒ F (gf ) and idF0 a → F1 (ida ),satisfying “obvious” coherence conditions.
This said, 2-category theory exists in many dialects: we mainly follow
a natural and auto-explicative notation based on the canonical reference[Kel89], but we feel free to diverge from it from time to time: the operationsof whiskering of an higher cell with a lower cell is denoted with the symbol ∗,so that F ∗ α : F H ⇒ F K and α ∗ G : HG ⇒ KG for natural transformation
α : H ⇒ K and suitable functors F, G.
Definition 7.2 : Let S : Aop × A → B be a strict 2-functor between strict
2-categories. A lax wedge based at S consists of a triple {b, ω Ob , ω hom }, where
17Obviously there are two possible choices for a direction in which a non-invertible“laxity cell” can go: a map endowed with a canonical 2-cell F (g)F (f ) ⇒ F (gf ); the same
correspondence, with 2-cells F (gf ) ⇒ F (g)F (f ) is called oplax functor. The reader willfind a personal mnemonic trick to remember how this nomenclature is chosen.

ENDS

67



b ∈ Ob(B) (the tip of the wedge) and collections of 1-cells ωa : b → S(a, a) ,
one for each a ∈ Ob(A), and 2-cells ωf : S(a, f ) ◦ ωa ⇒ S(f, a0 ) ◦ ωa0 , in a
diagramωaB

S(a, a)ωa

B

S(a, a)ωf 0
S(a,f )

ωf

ωa0

S(a,f 0 )

ωa0

S(f ,a )0

S(a,λ)

S(a,f )

0

S(a0 , a0 )S(a, a0 )S(a0 , a0 )S(f,a0 )S(a, a0 )S(f,a0 )These data must fit together in such a way that the following coherence
axioms, expressed by the commutation and pasting of the following diagramsof 2-cells, are satisfied:(1) The diagram of 2-cells

ωa00

S(a00 , a00 )S(f 0 ,a00 )

ωf 0

B
ωa0

S(a0 , a0 )S(a0 , a00 )S(f,a00 )

ωa00

ωa

S(a, a)ωf 0 fωa

S(a, a)ωf
S(a,f )

S(a, a0 )S(a, a00 )S(a00 , a00 )B

S(a,f 0 )S(a,f )

S(f 0 ,a00 )S(a0 , a00 )S(f,a00 )S(a, a0 )S(a, a00 )S(a,f 0 )is commutative for any λ : f ⇒ f 0 , i.e. the equation
ωf 0 ◦ (S(a, λ) ∗ ωa ) = (S(λ, a0 ) ∗ ωa0 ) ◦ ωf(137)

holds.
ff0(2) For each pair a −→ a0 −→ a00 of composable arrows in A, the diagramof 2-cells

68

F. LOREGIAN
ωa0
ωa0

B
σa

Θa

ωa

S(a0 , a0 )S(f,a0 )ωf

S(a, a)S(a,f )

B

σa0

σa

S(a, a0 )S(a0 , a0 )

Θa0S(f,a0 )σf

S(a, a)S(a, a0 )S(a,f )

is commutative, i.e. the equation(S(f, a00 ) ∗ ωf 0 ) ◦ (S(a, f 0 ) ∗ ωf ) = ωf 0 f

holds.
(3) For each a ∈ A, ωida = idωa .

(138)

Notation 7.3 : A lax wedge will be often denoted ω : b −→ S for short; this isevidently reminiscent of our Def. 1.5 and [ML98].
..

Definition 7.4 [Modification]: A modification Θ : ω V σ between two lax..
wedges ω, σ : b −→ S for S : Aop × A → B consists of a collection of 2-cells{Θa : ωa ⇒ σa }a∈Ob(A) such that the diagram of 2-cells
b0

b0

σa

σa

x

σa0

ωa

b
ωa0

S(a0 , a0 )S(a, a)ωf

S(f,a0 )S(a, a0 )σa0S(a, a)σf

S(a,f )

S(a,f )

S(a0 , a0 )S(f,a0 )S(a, a0 )is commutative, i.e.
(S(a, f ) ∗ Θa ) ◦ ωf = σf ◦ (S(f, a0 ) ∗ ωa0 )(139)

The definition of a modification is modeled on the definition of modification
between (lax, if necessary) natural transformations; modifications form the
3-cells of the 3-category 2-Cat, whose objects are 2-categories, 1-cells are (lax,if necessary) functors, 2-cells are (lax, if necessary) natural transformations.
Modifications follow rules for “whiskering” which are similar to those fornatural transformations and 2-cells, only in higher dimension (and in a muchcomplicated web of relations between all the possible compositions: there
are now three possible directions in which 3-cells can be composed!).

ENDS

69

Remark 7.5 : There is another more general definition for a modificationΘ : ω V σ between lax wedges having different domains, say {b, ω} and{b0 , σ}: it consists of a morphism ϕ : b → b0 and a 2-cell λa : σa ◦ ϕ ⇒ ωasuch that(σf ∗ ϕ) ◦ (S(a, f ) ∗ ma ) = (S(f, a0 ) ∗ ma0 ) ◦ ωf(140)
(draw the corresponding diagram of 2-cells!). Nevertheless, we are not interested in this alternative definition; Def. 7.4 entails that the set LFun(b, S) of..
lax wedges b −→ S is a category having morphisms precisely the modificationsΘ : ω V σ, and the correspondence βS : b 7→ LFun(b, S) is functorial. The
definition of lax end for S relies on the representability of this 2-functor.
Definition 7.6 [Lax end of S]: Let S : Aop × A → B bea lax
› a 2-functor;..
..
wedge ω : b −→ S is called the lax end of S, and denoted a S(a, a) −→ S if for
any other lax wedge σ : b0 → S there exists a single 1-cell x : b0 → b between
the tips of the wedges such thatωa ◦ x = σa ,ωf ∗ x = σfi.e. the diagram of 2-cellsE

hom(F a, Ga)

τa

τa 0

hom(F a0 , Ga0 )τf
F (f )∗

G(f )∗

hom(F a, Ga0 )commutes, and if every modification Θ : σ V σ 0 induces a unique 2-cell
λ : x ⇒ x0 (x0 is the arrow induced by σ 0 ) in such a way that λ ∗ ωa = Θa .
..
This ›realizes the isomorphism of categories between lax wedges b −→ S andB(b, a S(a, a)).
We denote, with an evident and harmless abuse of notation,“
BS = S(a, a).
(141)
a

›
Remark 7.7 : The notation chosen for a S has a meaning: ideally, the nco/end operation is depicted by an integral symbol (accordingly super- or
subscripted) overlapped by an 2n -agon;
in this way, a 2-end has the right to›be denoted as¸a “square-integral” , and a ∞-end (see Def. 7.31) should be
denoted as a symbol, the circle being a polygon with an infinite numberof sides.
7.1.1. Lax co/end calculus. Several kata of coend-fu proved in our §1
and §2 remain true after a proper “laxification”, justifying the intuition oflax co/ends as the right 2-categorical generalization of strict co/ends. We

70

F. LOREGIAN

collect the most notable examples of this phenomenon in the rest of thesection; the content of 7.1.2 surely deserves a special mention, as well asother remarks chosen to convey a sense of continuity and analogy. In 7.1.2
we prove that the lax counterpart of the ninja Yoneda lemma 2.1 provides
a reflection (using coends) and a reflection (using ends) of the category ofstrong presheaves into the category of lax presheaves.
Example 7.8 : The comma objects (f /g) [Gra74] of a 2-category B can be
identified with the lax end of functors 2op × 2 → B choosing the two 1-cells
f, g; this is a perfect analogy of Exercise 1.E10, in view of the characterizationof the comma object (f /g) as a lax pullback in Cat.
Example 7.9 : If F, G : A → B are 2-functors, then the lax end of the functor
is given by the formula“aB(F, G) : Aop × A → Cat(142)

B(F a, Ga) ∼= LFun(A, B)(F, G)(143)

where LFun(A, B)(F, G) is the set of lax natural transformations between
lax functors F, G : A → B defined in [Gra74].
Proof. A lax wedge for the 2-functor (a, a0 ) 7→ hom(F a, Ga0 ) amounts to asquare
h

a

a0
α

u

u0

c.

filled by a 2-cell τf : G(f )∗ ◦ τa ⇒ F (f )∗ ◦ τa0 . Each of the functors τa : E →hom(F a, Ga) sends e ∈ E into an object τa (e) such thatτfG(f ) ◦ τa (e) =⇒ τa0 (e) ◦ F (f )(144)

which is precisely what is needed to show thatthe correspondence e 7→Q{τa (e)}a∈A factors through LFun(A, B)(F, G) ⊆ a∈A hom(F a, Ga).


Lax natural transformations η : F 99K G, described as the lax end above,
can also be characterized as lax limits in the enriched sense: this motivatesthe search for a description of lax co/ends which is analogue to our 1.2,
where instead of strict co/equalizers we use [Kel89]’s notion of co/inserter.
For the ease of the discussion, we recall now how these lax limits aredefined (see [Kel89, §4]):

ENDS

71

Definition 7.10 : Let f, g : x → y be two parallel 1-cells in the 2-category C;
the inserter Ist(f, g) is defined as a pair (p, λ) where p : Ist(f, g) → x is a
1-cell in C and λ : f p ⇒ gp is a 2-cell, universal with respect to the property
of connecting f p, gp: this means that whenever we are given a diagram
x
f

q

y

µ

b
q

g

x

this can be split as the whiskeringqpbIst(f, g)

h

x
f

g

p
q

y

λ

x

for a unique 1-cell h : b → Ist(f, g) in C: this means, again, that ph = q andλ ∗ h = µ.

This remark motivates the search for a description of lax co/ends as lax
co/limits, on the same lines of our Remark 1.13; this is the content of Prop.
7.19 in this section.
Moreover, as an analogue of the claim for 1-dimensional co/ends, wewill prove that the lax co/end of a functor which is mute in a variablecoincides with the lax co/limit of the same functor restricted to the unmuted
component:
Remark 7.11 [Commutation of lax limits]: Strict limits are obviously
particular cases of lax limits; since the classical argument, slightly modified
to encompass the presence of non trivial laxity cells, applies to show thatlax limits commutes with lax limits (in an obvious sense which we invite thekeen reader to make precise), we obtain that lax co/limits (and lax co/ends)
commute with strict co/limits (and lax co/ends).
This simple remark will be used all along the present section, and in asimilar way we can deduce the “lax Fubini rule” for iterated lax co/ends:
here is an ordered exposition for these results.
Proposition 7.12 [Co/ends of mute functors]: Suppose that the 2-functor
S : Aop × A → B is mute in the contravariant variable, i.e. that there is a

72

F. LOREGIAN
S0

factorization S = S 0 ◦ p : Aop × A →− A −→ B“
0
S(a, a) ∼= q-lim
←−−S
p

(145)

a

hence every lax co/limit can be computed as a lax co/end.
Example 7.13 : As a particular example of this, if A is locally discrete (i.e.
identified with a locally small 1-category) andif the functor S 0 : A → B is›constant, i.e. S 0 (a) ≡ b for each a ∈ A, then a S is called cotensor of b by
A and is denoted b t A.
Theorem 7.14 [Parametric lax Ends]:
Whenever a functor F : Aop × A ×›
B→C› is defined, and the lax end a F (b, a, a) exists for every B ∈ B, thenb 7→ a F (b, a, a) extends to a 2-functor B → C which has the universal
property of the lax end of its mate F̂ : Aop × A → CB under the obvious
adjunction.
Theorem 7.15 [Fubini rule for lax co/ends]: If one among the following
lax ends exists, then so does the others, and the three are canonicallyisomorphic:
“
“ “
“ “


T (b, c, b, c)
T (b, c, b, c)
T (b, c, b, c)
(146)
b,c

b

c

c

b

Corollary 7.16 [Fubini rule for lax co/limits]: Lax limits commute: if
T : B × C → D is a 2-functor, we have
∼
q-lim
(147)
←−− q-lim←−− T (b, c) = q-lim←−− q-lim←−− T (b, c).
b∈B

c∈C

c∈C

b∈B

7.1.2. The lax ninja Yoneda lemma. The lax analogue of Prop. 2.1
acquires an extremely particular flavour in this context, since it is the gistof an argument which shows the co/reflectivity of the category of strict
presheaves Cop → Cat in the category of lax presheaves Cop 99K Cat: there
is a diagram of adjoint 2-functors
Fun(Cop , Cat)
(

)]

a a (

)[

LFun(Cop , Cat)This means that for each strict 2-functor H ∈ Fun(Cop , Cat) there are twonatural isomorphisms
Fun(Cop , Cat)(H, F [ ) ∼= LFun(Cop , Cat)(H, F ),∼ LFun(Cop , Cat)(F, H)
Fun(Cop , Cat)(F ] , H) =

(148)

ENDS

73

where the functors F [ and F ] are defined by the lax coends
“ a
“
[ ∼
F] ∼
C(
,
a)
×
F
a
F
=
= F aC(a, ) .

(149)

a

Proof. The proof exploits Example 7.9 as well as the commutation of co/ends
and lax co/ends, the preservation of lax co/ends by the hom functor, andthe strict ninja Yoneda lemma:ˆop[
Fun(C , Cat)(H, F ) = Cat(Hc, F [ c)ˆc“= Cat Hc, Cat(C(a, c), F a)aˆc “∼Cat(Hc, Cat(C(a, c), F a))
=
c a
“ ˆ
∼
Cat(Hc × C(a, c), F a)=“a c ˆ c∼
Hc × C(a, c), F a
= Cat
a
“
(7.9)
∼
= Cat(Ha, F a) = LFun(Cop , Cat)(H, F ).
a

The proof that Fun(C , Cat)(F ] , H) ∼= LFun(Cop , Cat)(F, H) is done in asimilar fashion.
op

Example 7.17 : Let 1 : Cop 99K Cat be the functor sending c ∈ C into
the terminal category, regarded as a lax functor. Then the strict functor1] : Cop → Cat is the lax coend
“ a
“ a
]
∼
∼
1 (c) =
1(a) × C(a, c) =C(a, c).
(150)›aHence the categoryC(a, c) coincides with the lax colimit of the strict
oppresheaf C → Cat, a 7→ C(a, c), which is [Str76, p. 171] the lax slice
category C//c of commutative diagrams of 2-cells
Bωa0

S(a0 , a0 )

ωaS(a, a)ωf

S(f,a0 )S(a,f )

S(a, a0 )A long and straightforward unwinding of universal properties shows that thecategory C//c enjoys the universal property of the lax colimit q-lim−−→C( , c).

74

F. LOREGIAN

Remark 7.18 [The twisted arrow category as a lax colimit]: Thisis an interesting remark. It is possible to characterize the twisted arrow
category of Def. 1.12 as the lax colimit of the diagram a 7→ A/a, i.e. as thelax coend
“ a
A/a.
(151)
This is another long and straightforward exercise in unwinding universal
properties, that we leave to the reader.
Proposition 7.19 : [Boz80, §2] There is a canonical isomorphism between thelax end of a 2-functor T : Cop × C → B and the limit of T weighted by thebifunctor C(( )] , ), i.e.
“C(( )] , ) ∼lim
T = T (a, a)
(152)
←−
a
›a
C(c, a) × C(a, c0 ) is the lax
where C(( )] , ) : (c, c0 ) 7→ C( , c0 )] (c) =composition of relators. A dual statement holds for lax coends.
Proof. The classical argument which exploits the conservativity of the Yoneda
embedding applies: we can computeB b, limC((←−

 (4.20)= limC(( ) , ) B(b, T )ˆ ←−](4.10) ∼B(b, T (c, d))C(c ,d)
=
c,d
ˆ
›a
=
B(b, T (c, d)) C(c,a)×C(a,d)
c,d
“ ˆ 
C(c,a)∼B(b, T (c, d))C(a,d)
=
a c,d
“ ˆ
ˆ


C(a,d)∼(7.15) =Cat C(c, a), B(b, T (c, d))
a c
d
“ ˆ
∼
Cat (C(c, a), B(b, T (c, a)))
=
a c
“
 “

∼
B(b,
T
(a,
a))
=
B
b,
T
(a,
a)
.
=

)] , )

T

a

]

a



We can define the tensor of a category T with an object a ∈ A, denoted
T · a by means of a lax coend; it is characterized by the natural isomorphism(in x)
Cat(T, A(a, x)) ∼(153)= A(T · a, x).
Now, let S : Aop → Cat, T : A → B be two functors and suppose B has
S×T·Cat-tensors; then the lax coend of the 2-functor Aop × A −−−→ Cat × B →
− B

ENDS

75

is called q-tensor product of S and T , denoted“ a
Sa · T a.
S ⊗ T =:

(154)

More fundamental and subtle results, like the reduction of co/ends toco/limits, and the preservation of co/ends by the hom functors, remain
valid for lax co/ends:
Theorem 7.20 : In a 2-category B, lax co/ends exist provided that B has
co/comma objects and Cat-co/limits.
Theorem 7.21 : The lax end of the functor T : Aop × A → B, if it exists, isuniquely determined by the natural isomorphism
 “
 “∼ B(x, T (a, a))B x, T (a, a) =
(155)
a

a

for every object x ∈ A. A dual statement holds for lax coends.

7.1.3. Applications: 2-distributors and lax Kan extensions. A 2distributor ϕ : A
B is a 2-functor ϕ : Bop × A → Cat. lax coends gives
a method to compose 2-distributors, as in the 1-dimensional case: more
precisely, let
A

ϕ

B

ψ

C

(156)

be a couple of composable 2-distributors, namely two 2-functors ϕ : B ×A →Cat and ψ : Cop × B → Cat; then the composition ψ  ϕ is defined by the
coend
“
op

ψ  ϕ(c, a) =b

ϕ(b, a) × ψ(c, b)(157)

The compatibility between lax colimits and products ensures that the expected associativity holds up to a canonical identification:(ω  ψ)  ϕ ∼(158)= ω  (ψ  ϕ)for any three A
BCD.
Let T : A → B and ϕ : A → C be two 2-functors.
ϕψωDefinition 7.22 : We call (left) quasi-Kan extension of ϕ along T a 2-functor
LanT ϕ : B → C endowed with a quasi-natural transformation a : ϕ ⇒LanT ϕ ◦ T (a unit) such that, for each 2-functor S : B → C endowed with
λ : ϕ ⇒ S ◦ T there exists a unique Cat-natural transformation ζ : LanT ϕ ⇒S such that(ζ ∗ T ) ◦ α = λ(159)

and moreover, if Σ : λ V λ0 is a modification between quasi-natural transformations, there is a unique modification Ω : ζ V ζ 0 (where ζ is induced by λ,76

F. LOREGIANand ζ 0 by λ0 ) between Cat-natural transformations such that (Ω ∗ T ) ◦ α = Σ.
This can be expressed with the isomorphismhih∼ CB LanT ϕ, S
Fun(A, C) ϕ, S ◦ T =which is natural in S.

i

(160)

Example 7.23 : The quasi-Kan extension of a 2-functor ϕ : A → C along thetrivial 2-functor A → 1 is the lax colimit of ϕ.

Remark 7.24 : We can obtain different notions of quasi-Kan extension by
reversing the directions of α, λ, ζ etc.
The example above, as well as the following theorem, shows that the choiceof Cat-natural transformations instead of quasi-natural transformations is
the right choice (see also [Boz80] for a dual statement):
Theorem 7.25 : Notations as above. If C has at least tensors B(T a, b) · ϕa0
for each a, a0 ∈ A, and the lax coends“ a
B(T a, b) · ϕa(161)then the quasi-Kan extension of ϕ along T exists, and it is given by theformula above.
We can mimick also Exercise 2.E1 to obtain a lax analogue of it:
Proposition 7.26 : Let LNat(U, V ) denote the category of lax natural transformations between two 2-functors U, V . Then
LNat(F × G, H) ∼(162)
= LNat(F, H G ),›where H G (x) = LNat(よ(A) × G, H) = y Sets(hom(y, x) × Gy, Hy)

Proof. It is a computation in coend-fu, and every step can be motivated byresults in the present section: calculemus.
“
G
LNat(F, H ) = Sets(F x, LNat(よ(A) × G, H))“x “∼
Sets(F x, Sets(hom(y, x) × Gy, Hy))=x y“
 “ x
∼F x × hom(y, x) × Gy, Hy= Setsy“∼= Sets F y × Gy, Hyy= LNat(F × G, H).



ENDS

77

7.2. Homotopy coends and ∞-coends. Higher category theory is now
living a Renaissance, thanks to a massive collaboration of several people
drawing from various fields of research, and cooperating to re-analyze every
feature of category theory in the topos of simplicial sets.
Several reasons, and the urge to keep this chapter finite-dimensional
force us to take for granted a certain acquaintance with the language of∞-categories à la Joyal-Lurie, but we also try to offer at least an intuition
for what’s going on and why things are done that way. The reader seeking
a deeper understanding of this topic is advised to quit their job and move
to Mojave desert with [Lur09, Joy08, Joy02a, JT07] and a bag full of good
weed.
Here we present the theory of homotopy co/ends in model category theory,and then we move to the theory of simplicially coherent and quasicategoricalco/end calculus. A final paragraph explores the definition of a co/end
in a derivator, and this concludes the discussion of co/end calculus ineach of the most common models for higher categories (model categories,
enriched categories, simplicial sets, derivators). We leave aside a rather
important question, that is model dependence: for example, does an homotopy
co/end correspond to an ∞-co/end if we pass from model categories toquasicategories?
In our discussion we follow the unique references available: [Isa09] forco/ends in model categories, [CP97] for sSet-enriched co/ends, and [GHN15]
for quasicategorical ones. The definition of a coend in a derivator comesfrom the extensive treatment of category theory in Grothendieck’s derivators
started by M. Groth [Gro13].
7.2.1. Co/ends in model categories. One of the most important partsof model category theory (the study of those structures that set homotopy
theory in a purely formal framework) is the study of homotopy co/limits.
It is a fact, inherent to the theory, that colimit functors limJ : CJ → Care−→often quite ill-behaved with respect to a homotopical structure: such athing is determined by the specification of a distinguished class of arrows
W ⊆ hom(C) (these are called weak equivalences) which is the class of
isomorphisms in a “localization” of C. If C has such a structure, then every
category CJ acquires an analogous structure WJ where η : F ⇒ G is in WJ
if and only if each component ηj : F j → Gj is in W.
It is a fact that the image of such a natural transformation η : F ⇒ G under
the colimit functor, lim η : lim F → lim G is not always a weak equivalence.18−→−→−→
18A minimal instructive example goes as follows: take J to be the generic span 1 ← 0 → 2

and the functor sending it to ∗ ← S n−1 → ∗; the colimit of F is the terminal space ∗. We
can replace F with the diagram D2 ← S n−1 → D2 , and since disks are contractible there

78

F. LOREGIANThis is an unavoidable feature of the colimit functor limJ : CJ → C. One−→of the main tenets of homotopy theory is, nevertheless, that it doesn’t matter
if we replace an object with another, as soon as the two yield equivalent
results. There is hope, then, that the category of functors CJ contains abetter-behaved representative for the functor lim, and that the two are linked−→
by some sort of weak- or homotopy equivalence.
That’s what a homotopy colimit is: a deformation holimthat−−−→ of lim−→
preserves pointwise weak equivalences. And this is a general procedure inhomotopy theory, where most objects X are not “compatible” with thehomotopical structures we superimpose on our category of spaces, and yet
one is often able to find better-behaved representatives X̃ in the same
homotopy class.
This said, there are mainly two ways to link co/end calculus and homotopy
theory:• In nice situations, homotopy co/limits can be computed as co/ends:
the first attempt to clarify this construction was given in [BK72]; niceexplanatory surveys about this theory (touching also homologicalalgebra) are [Hör14]´ and [Gam10].
colimit)• The co/end functor : Cat(Cop × C, D) → D (as a particular´
can be “derived” yielding an homotopy co/end functor : Cat(Cop ×C, D) → D that preserves weak equivalences. This perspective, which
is not independent from [CP97]’s point of view, is expanded in [DF78]
and [Isa09].
These are respectively a co/end calculus applied to model categories andinterpreted in model categories. The interplay and mutual completion ofthese two perspectives is evident.
Remark 7.27 : Let  : A × B → C be the t part of a thc situation (see 3.13),
which is moreover left Quillen, and let I be a Reedy category ([Hov99, Def.
5.2.1]). Then the coend functor
ˆ: Cat(Iop , A) × Cat(I, B) → C(163)
is a left Quillen bifunctor if we regard the functor categories Cat(Iop , A)
and Cat(I, B) endowed with the Reedy model structure.
7.2.2. Co/ends in quasicategories.
Remark 7.28 : As a rule of thumb, the translation procedure from category
to ∞-category theory is based on the following meta-principle: first you
rephrase the old definition in a “simplicially meaningful” way, so that theis a homotopy equivalence F̃ ⇒ F ; unfortunately, the induced arrow lim F̃ = S 2 → ∗ is−→
not a weak equivalence.

ENDS

79

∞-categorical definition specializes to the old one for quasicategories N (C)which arise as nerves of categories. Then you forget about the original gadgetand keep the simplicial one; this turns out to be the right definition.
The first victim of this procedure is the twisted arrow category 1.12 of an∞-category.

Definition 7.29 [Twisted arrow ∞-category]: Let ε : ∆ → ∆ be thefunctor [n] 7→ [n] ? [n]op , where ? is the join of simplicial sets [Joy08, EP08].
Let C be a ∞-category; the twisted arrow category tw(C) is defined to be
the simplicial set ε∗ C, where ε∗ : sSet → sSet is the induced functor. Moreexplicitly, and consequently, the n-simplices of tw(C) are characterized by
the relation
tw(C)n ∼(164)= sSet(∆[n], tw(C)) ∼= sSet(∆[n] ? ∆[n]op , C).

The most important feature of the twisted arrow category is that it admits
a fibration over Cop × C (part of its essential properties can be deduced
from this); the machinery of left and right fibrations exposed in [Lur09, Def.
2.0.0.3] gives that
• There is a canonical simplicial map Σ : tw(C) → Cop × C (inducedby the two join inclusions ∆[ ], ∆[ ]op → ∆[ ] ? ∆[ ]op );• This ∞-functor is a right fibration in the sense of [Lur09, Def. 2.0.0.3].
Remark 7.30 : It is rather easy to see that a 0-simplex in tw(C) is an edgef : ∆[1] → C, and a 1-simplex of tw(C) is a 3-simplex thereof, that we candepict as a pair of edges (u, v), such that the square having twisted edges
(165)

o u
f0

f



v

/

commutes. This suggest (as it must be) that the definition of tw(C) for a∞-category specializes to the 1-dimensional one and adds higher-dimensional
informations to it.
Definition 7.31 : Let C, D be two ∞-categories; the co/end of a simplicial
map F : Cop × C → D is the co/limit of the composition→D
tw(C) −→ Cop × C −
Σ

F

(166)

The main interest of the authors in [GHN15] is to formulate an analogue
of 4.5, which characterizes the Grothendieck construction of a Cat-valued
functor as a particular weighted colimit (see 4.22).
It is rather easy to formulate such an analogue definition: this appears as[GHN15, Def. 2.8].

80

F. LOREGIAN

Definition 7.32 [op/lax colimit of F ]: Let F : C → Cat∞ be a functor
between ∞-categories. We define• the lax colimit of F to be the coend
˛ c
Cc/ × F c(167)
• the oplax colimit of F to be the coend
˛ c
C/c × F c

(168)

The Grothendieck construction associated to F , discussed in [Lur09] withthe formalism of un/straightening functors results precisely as the oplax
colimit of F . This is coherent with our 4.22 and 4.23.
7.3. Simplicially coherent co/ends. All the material in the following
subsection comes from [CP97]. Since we are forced to divert from [CP97]’s
notation by our personal choices and a slight pedantry, we begin the exposition establishing a convenient notation and a series of useful short-hands.
We decided to keep this introduction equally self-contained and simple, but
we can’t help but admit that• there is a sheer amount of (unavoidable, and yet annoying) sins ofomissions in this survey, basically due to the ignorance of the author;
moreover, the price to pay to obtain a self-contained exposition is todeliberately ignore several subtleties, exposing the theory to several
simplifications.
• there are newer and more systematic approaches to this topic, owing
a great debt to [CP97] but capable to generalize sensibly their constructions; among many, the reader should consult the exceptionally
clear [Rie14, Shu06]. All these references reduce the construction ofa simplicially coherent co/end to the “unreasonably effective co/bar
construction” [Rie14, Ch. 4].
It is our sincere hope that this does not affect the outreach of this elegant
but neglected flavour in which to do higher category theory, and the clumsy
attempt to popularize an account of [CP97] has to be seen as an attempt
to communicate how beautiful we find this writing, as it is (one of) the
beginner(s) of categorical homotopy theory.
Local notation. All categories A, B, . . . appearing in this subsection are∆op , Sets]. All of them possess the co/tensors (seeenriched over sSet = [∆Def. 2.3) needed to state definitions and perform computations. These
functors assemble into a thc situation (see Remark 3.13) t = (·, hom, t)
where · : sSet × A → B determines the variance of the other two functors. A
useful shorthand to denote the functor t (K, A) = K t A (especially when

ENDS

81

it is necessary to save space or invoke the “exponential” behaviour of thisoperation) is AK . We switch rather freely among these two notations.
Definition 7.33 [Totalization]: Let Y • : ∆ op × ∆ → B be a simplicialcosimplicial object; we define the totalization tot(Y•• ) of Y • to be the end
•ˆ∆[n] t Y n
(169)
∆
n∈∆

(it is a cosimplicial object). The totalization of Y • is also denoted with theshorthand ∆ • t Y • or similar.

Definition 7.34 [Diagonalization]: Let X• : ∆ op × ∆ op → B be a bisimplicial object; we define the diagonalization diag(X• ) of X• to be the coend∆ ˆ n∈∆
∆[n] · Xn(170)•(it is a simplicial object) The diagonalization of X• is also denoted with theshorthand ∆ • · X• or similar.

Notation 7.35 [Chain co/product]: Let A ∈ sSet-Cat, and ~xn = (x0 , . . . , xn )the “generic n-tuple of objects” in A; given additional objects a, b ∈ A definea bisimplicial set
q A[a|~xn |b]• :=ax0 ,...,xn ∈AA(a, x0 ) × A(x0 , x1 ) × · · · × A(xn , b).19(171)

Faces and degeneracies are induced, respectively, by composition and identityinsertion (see Exercise 7.E4).
Finally we define the simplicial set δA(a, b) to be diag(qA[a|~x• |b]). Couched
as a coend, δA(a, b) is written
ˆ n∈∆
∆
∼
δA(a, b) =∆[n] × qA[a|~xn |b]
=
∼
=

ˆ n∈∆
∆
ˆ n∈∆
∆

∆[n] ×aax0 ,...,xn ∈A

x0 ,...,xn ∈AA(a, x0 ) × A(x0 , x1 ) × · · · × A(xn , b)∆[n] × A(a, x0 ) × A(x0 , x1 ) × · · · × A(xn , b)

19It is useful to extend this notation in a straightforward way: A[a|~x|b] denotes the

product A(a, x0 )×A(x0 , x1 )×· · ·×A(xn , b), and ΠA[a|~x|b], A[~x], ΠA[~x], qA[~x] are definedsimilarly. Note that qA[a|~xn |b]• does not depend on ~xn since the coproduct is quantified
over all such ~xn ’s.

82

F. LOREGIAN

Definition 7.36 [The functors Y and W]: Let T : Aop × A → B be afunctor; we define Y(T )• to be the cosimplicial object (in B)
Y(T )n :=YA[~xn ] t T (x0 , xn )



(172)

a

A[~xn ] · T (x0 , xn ).

(173)

~
x=(x0 ,...,xn )where A[~xn ] = A(x, x1 ) × · · · × A(xn−1 , xn ). Dually, given the same T , wedefine W(T )n to be the simplicial object (in B)
W(T )n :=~x=(x0 ,...,xn )Example 7.37 : If we consider A to be trivially enriched (or as someone says,
a discrete simplicial category), then the object A[a|~xn |b]• coincides with thenerve of the category (a ↓ A ↓ b) of arrows “under a and above b”.

Definition 7.38 [Simplicially coherent co/end]: Let T : Aop × A → B
be a functor. We define˛ˆ˛ aˆ a0 ,a00
0 00 δA(a0 ,a00 )T (a, a) :=T (a , a )T (a, a) :=δA(a0 , a00 )·T (a0 , a00 )a

a0 ,a00

to be the simplicially coherent co/end of T .

(174)

In a few words, the definition of a sSet-coherent co/end involves theclassical construction but adds to the scene “fattened up” co/tensors A[a|~x|b]and suchlike, organized as a functor Aop × A × Aop × A → B in such a waythat the co/end is on two variables a0 , a00 ∈ A.

Remark 7.39 [ , coherent co/ends as deformations]: Example 4.16
gives that co/end are weighted co/limits, and precisely weighted co/limits
with the hom weight.
This perspective
is useful here, as in some sense we¸´
0 00
are writing that a T is the end (a0 ,a00 )∈Aop ×A T (a0 , a00 )hom(a ,a ) where weapplied a suitable “deformation” (or “resolution”, or “replacement”) functor
δ to the hom functor A( , ), seen as the identity relator (Remark 5.7). To
some extent this point of view is explored in [Gen15] in the particular casewhere V-Cat = dg-Cat; we have taken something from, as well as given
something to, that document.
This perspective is of great importance to encompass coherent co/ends
into a general theory “compatible” with some model structure on V-Cat, forsome monoidal model V and the Bousfield-Kan model structure on V-Cat.
Expanding this point of view, will be, hopefully, the subject of [GL].
Homotopy coherent calculus. Classical co/end calculus (in the triptych
Fubini - Yoneda - Kan) is an invaluable tool (in fact, the only and mostnatural one) to prove several results even in the simplicial setting: as it iscustomary in the salons of higher category theory, we will always reduce a

ENDS

83

computation involving a coherent, and yet indirectly defined object, to acomputation involving a “concrete”, and yet complicated object which takes
into account the coherence introduced in the deformation.
[CP97] succeed in the indeed quite ambitious task to to rewrite all themost important pieces of classical category theory in this “higher” model(the paper contains a calculus for co/limits, mapping spaces, Yoneda, andKan extensions). The aim of the rest of this subsection is to sketch someof these original definitions, hopefully helping an alternative formulation of(∞, 1)-category theory (the authors of whom we owe a great debt, were it
only for having proved –in 1997!– that the “∞-categorical dogma” can be
avoided20) to escape oblivion: we do not claim to give a complete account of
this theory here, and instead address the interested reader to their beautiful
original paper.
Proposition 7.40 : Let T : Aop × A → B be a sSet-functor. Then there is a
canonical isomorphism
˛
T (a, a) ∼(175)
= tot(Y(T )• )a

Proof. We use heavily the ninja Yoneda lemma 2.1 in its enriched form,whereˆ
∼ F (B)
A(X, B) t F (X) =(176)
X

and the fact that K t (H t A) ∼= (K ⊗ H) t A, naturally in all arguments.
With this remark in hand we can move to the real proof: ~x = (x0 , . . . , xn )is a generic tuples of objects of A, and to save some space we switch to thenotation AK to denote K t A.
˛ˆ
0 00
T (a, a) :=T (a0 , a00 )δA(a ,a )aa0 ,a00ˆ´
000∼
T (a0 , a00 ) n ∆[n]×qA[a |~x|a ]=a0 ,a00
ˆ0
00∼
T (a0 , a00 )∆[n]×qA[a |~x|a ]=000
a ,a ,nˆ∆[n]×qA[a0 |~x]00∼
T (a0 , a00 )A(xn ,a )=a0 ,a00 ,nˆ∆[n]×A[a0 |~x]
Y 00∼
T (a0 , a00 )A(xn ,a )=a0 ,a00 ,n x0 ,...,xn

20The subtle monophysism called “∞-categorical dogma” asserts that ‘quasicategory’and ‘∞-category’ are synonyms.

84

F. LOREGIAN

Y

∼
=

ˆ

∼
=

ˆ

∼
=

ˆ

∼
=

ˆ  Y

a0 ,n x0 ,...,xnY

n x0 ,...,xnn00 A(xn ,a00 )T (a , a )0a00

∆[n]×A[a0 |~x]

T (a0 , xn )∆[n]×A[a |~x]0a0 ,n x0 ,...,xn

Y

ˆ

ˆ

a0

A(a0 ,x0 )T (a , xn )0

T (x0 , xn )A[~xn ]x0 ,...,xn∆[n]∆[n]×A[~x]∼ tot(Y(T )).
=

For¸ the sake of completeness, we notice that the universal wedge testifyingthat a T (a, a) ∼= tot(Y(T )) is induced by the morphisms¸

a

T (a, a) =´

a0 ,a00

δA(a0 , a00 ) t T (a0 , a00 )o´

a0 ,a00 ,n∆[n] × qA[a0 |~x|a00 ] t T (a0 , a00 )∆[n] t Y(T )n .

Prove the dual statement as an exercise (to finish the proof it is of vital
importance to exploit a good notation):
Proposition 7.41 : Let T : Aop × A → B be a sSet-functor. Then there is a
canonical isomorphism
˛ a
T (a, a) ∼(177)
= diag(W(T )• )7.3.1. Simplicially coherent natural transformations.
Remark 7.42 : The homotopy coherent co/ends admit “comparison” maps tothe classical co/ends; this is part of a general tenet of higher category theory,
where homotopically correct objects result as a deformation of classical ones,
and this deformations maps
¸ into/out´of the classical object.
The comparison map T (a, a) → T (a, a) arises, here, as an homotopy
equivalence between the simplicial set A(a, b) (seen as bisimplicial, andconstant in one direction) and the bisimplicial set δA(a, b) = diagA[a| • |b]• :this is [CP97, p. 15].
The mapad0 :A(a, x0 ) × A(x0 , b) → A(a, b)
(178)
x0

ENDS

85

given by composition has an homotopy inverse given bys−1 : A(a, b) → A(a, a) × A(a, b) : g 7→ (ida , g).

(179)

Indeed, the composition d0 s−1 is the identity on A(a, b), whereas the composition s−1 d0 admits is homotopic to the identity on δA(a, b) (we use the same
name for the maps d0 , s−1 and the induced maps d¯0 : δA(a, b) → A(a, b),
induced by the universal property, and s̄−1 : A(a, b) → δA(a, b)).
There is an important difference between these two maps, though: whilst
d0 is natural in both arguments, s1 is natural in B but not in A. This has an
immediate drawback: whilst d0 can be obtained canonically, as the universal
arrow associated to a certain natural isomorphism (see (183) below), s−1
can’t (the best we can do is to characterize the natural argument of s−1 via[CP97, Example 2, p. 16]).
As we have seen in 1.19, the set of natural transformations
between two´
functors F, G : C → D coincides with the end x D(F x, Gx), and (see 7.9)
the category of lax ›natural transformations between two 2-functors coincides
with the lax end x D(F x, Gx). It comes as no surprise, then, that thefollowing characterization of homotopy coherent natural transformations
between two simplicial functors hold:Definition 7.43 [Coherent natural transformations]: Let F, G : C → D
be two simplicial functors; then the simplicial set of coherent transformations
between F and G is defined to be
˛
Coh(F, G) := D(F a, Ga).
(180)
aDefinition 7.44 [Mean tensor and cotensor]: Let F : A → B, G : A →sSet, H : Aop → sSet. We define G t F , H ⊗ F respectively as˛
˛ a
G t F := Ga t F a
H ⊗ F :=Ha ⊗ F a.
(181)
a

Definition 7.45 [Standard resolutions]: Let F : A → B be a simplicial
functor; we define˛F a := A(a, ) t F = A(a, x) t F x
˛x x
F a := A( , a) ⊗ F =A(x, a) ⊗ F x.
Example 7.46 : We specialize the above definition to compute the functors
hom(a, ) and hom(a, ): in particular we concentrate on the second case,since the first is completely dual.
˛ aA(a, x) × A(x, b)hom(a, b) =

86

F. LOREGIAN

∼
=

ˆ xy

δA(x, y) × A(a, x) × A(y, b)

∼
=

ˆ xy

A(a, x) × δA(x, y) × A(y, b)

∼
=

ˆ xyn

∼
=

ˆ xyn

a

x0 ,...,xn

A(a, x) × A(x, x0 ) × · · · × A(xn , y) × A(y, b) × ∆[n]A[a|x̃n |b] × ∆[n] ∼= δA(a, b).

We leave as an easy exercise in co/end-fu the proof of the following result(see Exercise E5), which shows that the standard resolutions ( ), ( ) ‘absorb
the coherence informations’:
Proposition 7.47 : Let F, G : C → D be two simplicial functors; then there
are canonical isomorphisms∼ Coh(F, G) ∼(182)
Nat(F, G) == Nat(F , G).

This result has a number of pleasant consequences: the simplicially coherent setting is powerful enough to retrieve several classical constructions.
• Example 7.46 above shows that hom(a, )(b) ∼= δA(a, b); this entails
that there is an isomorphism
Nat(δA(a, ), A(a, )) ∼(183)
= Coh(A(a, ), A(a, ))and it is a matter of verifying some additional nonsense to see that thesSet-natural transformation corresponding to the identity coherent
transformation is precisely d0 .
• The map d0 defines additional universal maps ηF , η F which “resolve”
a functor F : A → B whenever F , F exist (it is sufficient that B
admits all the relevant co/limits to perform the construction of
F , F ). From the chain of isomorphisms˛η F : F b = A(b, a) t F aˆa∼δA(a0 , a00 ) t A(b, a00 ) t F a0=
a0 ,a00ˆ←A(a0 , a00 ) t A(b, a00 ) t F a0
a0 ,a00(2.1) ∼
= F b;˛ a
ηF : F b =F a ⊗ A(a, b)∼=ˆ a0 ,a00

F a0 ⊗ A(a00 , b)δA(a0 , a00 )

ENDS

→

ˆ a0 ,a00∼ F b;=

87

F a0 ⊗ A(a00 , b)A(a0 , a00 )we obtain natural transformations corresponding to suitable coherent
identities under the isomorphism of Prop. 7.47.
• The maps ηF , η F behave like resolutions: [CP97, Prop. 3.4] showsthat they are level-wise homotopy equivalences (meaning that ηF : F a →(ηF )∗

F a induces homotopy equivalences of simplicial sets B(b, F a) −−−→
B(b, F a) for each b, naturally in b).21

7.3.2. Simplicially coherent Kan extensions. The universal property
characterizing a Kan extension is inherently 2-dimensional: uniqueness isstated at the level of 2-cells, and any sensible generalization of it to thehigher world involves a “space” of 2-cells between 1-cells. This entails thatany reasonable definition of a (left or right) Kan extension ultimately relies
on a nice definition for a space of coherent natural transformations between
functors, which has been the subject of the previous subsection. There are,nevertheless, several subtleties as there are many choices available for adefinition: in the words of [CP97],
Clearly one can replace Nat by Coh [in the definition of a Kan
extension], but should isomorphism be replaced by homotopy
equivalence, should this be natural, in which direction should
this go. . . ?
As it turns out from [CP97], the right way to preserve a reasonably vastcalculus for Kan extensions is to ask that the isomorphisms
Nat(H, RanG K) ∼= Coh(HG, K)
Nat(LanG H, K) ∼= Coh(H, KG)
hold. This can be achieved defining the left and right Kan extensions asfollows:
Definition 7.48 [Coherent Kan extensions]: Let F : A → C and G : A →B be a span of simplicial functors; we define
˛
RanG F ( ) = B( , Ga) t F a
˛ aa
LanG F ( ) =B( , Ga) ⊗ F a
21We decide to skip the proof of this proposition, as it is quite long, technical, andeven though it relies on co/end-fu it doesn’t add much to the present discussion.

88

F. LOREGIAN

Remark 7.49 : This can be seen as a simplicially coherent analogue of our
7.25; it is not a coincidence that lax and simplicially coherent co/end calculi
mimick each other: 2-co/ends correspond to suitable “truncated” simpliciallycoherent co/ends (and this correspondence can be made functorial).
In the same spirit of [Boz80], a co/endy view on categorical homotopy
theory sheds a light on several geometric constructions (see [CP97] for more
informations and links with [CP90, Seg74]).
We would like to prove, now, that the isomorphisms defining a Kan
extension hold with the definitions above. This is a computation in co/endfu, which at this point can be left as an exercise for the reader.
7.3.3. Co/ends in a derivator. The theory of derivators serves as apurely 2-categorical model for higher category theory, where all the coherence
informations are encoded in coherence conditions for suitable diagrams of
2-cells. Here we only sketch some of the basic definitions needed to pave theway to Def. 7.51 below.
Definition 7.50 [The 2-category PDer]: A prederivator is a strict 2-functor
D : Catop → CAT (where CAT is the category of + -categories, see the twouniverse convention in the introduction); a morphism of prederivators isa pseudonatural transformation between pseudofunctors, η : D ⇒ D0 ; a2-cell between morphisms of prederivators is a modification (see Def. 7.4)
Θ : η V η 0 between pseudonatural transformations.
These data form the 2-category of prederivators.
Ω

The notion of a derivator arises as a refinement of this; apart fromsome minor (milder, but not less important) assumptions, a derivator is aprederivator D such that every D(u) : D(J) → D(I), induced by u : I → J
has both a left and a right adjoint, fitting into a triple
u! a u∗ a u∗ :

D(J)

u!
u∗

D(I)

u∗

(see [Gro13, Def. 1.10]). These functors are called respectively the homotopy
left and right Kan extensions along u : I → J. Axiom (Der4) in [Gro13,
Def. 1.10] states that these Kan extensions can always be computed witha pointwise formula; this can be interpreted as a rephrasing of the theory
exposed in our 2.1, in view of the equivalence between all the following
characterizations of LanG F (b)/RanG F (b):• the (right or left) Kan extension of F : A → C along G : A → B
computed in b;• the weighted co/limit of F with respect to the representable hom(G, b);• the conical co/limit of F over the category of elements of hom(G, b);

ENDS

89

• the conical colimit of the diagram (G ↓ b) → A −→ B.
Let Tw(K) be the category of elements (Def. 4.1) of homK , for a small
category K; then there exists a functor ΣK = (t, s) : Tw(K) → K op × K(Prop. 4.5).
F

Definition 7.51 [Homotopy coend in a derivator]: Let D be a derivator,´K
and K ∈ Cat a category. The homotopy coend: D(J × K op × K) → D(J)is defined as the composition
´K

Σ∗

p

!
: D(J × K op × K) −−K→ D(J × Tw(K)) −
→
D(J)

(184)

Remark 7.52 : Let D(J| ) : Catop → CAT be the shifted derivator of D, i.e.
´K
D( )
×J
the functor I 7−→ I × J 7→ D(I × J). Then the coend
is a morphism
between the shifted derivators D(K op × K| ) → D( ).
Remark 7.53 : There is obviously a similar notion of homotopy end in D:
one only has to replace p! with the right adjoint p∗ in the definition above(taking the limit over the twisted arrow category, instead of the colimit):Σ∗p∗
op × K) −−K→ D(J × Tw(K)) −→D(J)K : D(J × K

´

(185)

Lemma 7.54 : If F : D → D0 is a morphism of derivators, there is a canonical“comparison” morphism´K´K◦F → F ◦(186)
obtained as the composition

pr! ◦(t,s)∗ ◦FJ×L×K op ×Ko

D(J × K × K op )pr! ◦FJ×L×Tw(K) ◦(t,s)∗

D0 (J)FJ×L ◦pr! ◦(t,s)∗

where the second morphism results as the bc pasting
p! FJ η(p ap∗ )p ϕJ p(p ap∗ ) Fe p!

!
!
p! FJ =======⇒ p! FJ p∗ p! ==! ==⇒! p! p∗ Fe p! =======⇒ F ep!

represented in the following diagram of 2-cells:

(187)

90

F. LOREGIANp!

D0 (e)id

D0 (J)FJp∗
(pr! apr∗ )D0 (e)

D(J)
p∗

Fe

D(e)

id
η(pr! apr∗ )
p!

D(J)

It is almost a triviality that a derivator morphism F preserves homotopy
coends (i.e. the above 2-cell is invertible) if it preserves colimits, or moregenerally Kan extensions.
Exercises for §7
E1 A lax colimit for a diagram F : J → K in a 2-category K is an object L with alax cocone {F j → L} satisfying a suitable universal property (state it, mimicking–the dual of– Def. 7.2). Show that the opposite of the twisted arrow category
tw(C) of Def. 1.12 is the lax colimit of the diagram C → Cat : c 7→ C/c sending
every object to its slice category.
..
→ d for a 2-functor S : Aop × A → B; state
E2 State the definition of lax cowedge S −the definition of lax coend for S as an initial cowedge, the representing object of
the functor d 7→ LCwd(S, d).
E3 Show that Ist(f, g) share the universal property of the Cat-limit of the diagram{0 ⇒ 1} → C choosing f, g weighted by the Cat-presheaf {0 ⇒ 1} → Catd0choosing the categories {0} ⇒ {0 < 1}.
d1

is
E4 Define co/faces and co/degeneracies for the objects Y(T )and W(T ) (hint: there ∼ A[x1 |~an isomorphism τ : ΠA[~x] t T (x0 , xn ) =y |xn ]n−1 A(x0 , x1 ) t T (x0 , xn ) ,and you want to assemble a map Y(T )n−1 → Y(T )n from its components
ΠA[~x] t T (x1 , xn ) → Y(T )n ; this defines d0 . The map dn is defined via an
isomorphism σ and a similar argument).
E5 Prove that Nat(F, G) ∼= Coh(F, G) ∼= Nat(F , G), using Def. 7.38 and a formal
argument.
E6 Prove that the isomorphisms
Nat(H, RanG K) ∼= Coh(HG, K)
Nat(LanG H, K) ∼= Coh(H, KG)

hold defining
Kan extensions as in 7.48.
´ Kcoherent
E7 Prove that
: D(K op × K|−) → D defines a morphism of derivators (you caneither prove that a functor u : K → L induces a morphism between the shifted
derivators D(L|−) → D(J|−), or prefer an explicit argument –both ways are
considerably long).
E8 Prove that “coends in a derivator are pointwise”,
given an arrow´ K i.e. that´ K
j : e → J there is a canonical isomorphism j ∗X ∼j ∗ X for each=X ∈ D(J × K op × K).
E9 State and prove the Fubini theorem for homotopy coends in D: the diagram

ENDS

91

Σ∗
K

D(J × K oo × Loo )D(J × Tw(K) × Loo )

Σ∗
L

prK
!

α

D(J × K oo × Tw(L))

D(J × Loo )

ΣK×L

prL
!
oo

D(J × K )

Σ∗L

D(J × Tw(L))

β

Σ∗
K

prL
!

D(J × Tw(K))

D(J)

prK
!

commutes for canonically determined 2-cells α and β.
E10 State and prove an existence theorem for weighted colimits in a derivator: given abimorphism  : (DSets , D) → D(I|−), we define the colimit of X ∈ D(J), weighted´J
by W ∈ DSets as the coend (in D)
W  X, i.e. as the image of the pair (W, X)
under the composition´J

DSets (J op ) × D(J) −→ D(I|J op × J) −−→ D(I).
·

(188)

Appendix A. Promonoidal categories
A promonoidal category is what we obtain taking the definition of a
monoidal category and we replace every occurrence of the word functor
with the word profunctor (here we refrain to adopt the name “relator” sincerel-monoidal category is a terrible name).
More precisely, we define
Definition A.1 [Promonoidal structure on a category]: A promonoidal
category consists of a category C which is a monoid object in Relt, the category of profunctors defined in 5.1.
A monoid object in Relt is a category C, endowed with a bi-profunctor
P: C×CC (the monoidal multiplication) and a profunctor J : 1
C(the monoidal unit), such that the following two diagrams are filled by theindicated 2-cells (respectively, the associator and left/right unitor) in Relt:
P ×hom

C×C×C

hom ×P _



C×C

px

α


P

/ C×C
_P


/C

C

J×hom

ρ
hom




/ C × C o hom ×J
_P


C

λ



C

hom

(189)
These data form what is called a promonoidal structure on the category C,
denoted P = (P, J, α, ρ, λ).
Remark A.2 : Coend calculus allows us to turn the conditions
P  (P × hom) ∼= P  (hom ×P )

(190)

92

F. LOREGIANP  (J × hom) ∼= hom ∼
= P  (hom ×J)(191)

giving the associativity and unit of the promonoidal structure into explicit
relations involving the functors P : Cop ×Cop ×C → Sets and J : C → Sets;22
we have the following rules, written in Einstein notation:• The associativity condition for P : C×C
C amounts to saying thatthe following boxed sets, obtained as coends, are naturally isomorphic(via a natural transformation αabc;d having four components, threecontravariant and one covariant).
´ xy xy x bc
(P  (hom ×P ))abc;d =Pd Ha Py´ y ´ x xy x  bc∼z
P d Ha P y=´∼ z P ay P bc=yd
´ xy xy c ab∼
(P  (P × hom))abc;d =Pd Hy Px
´ z ab xc∼
Px Pd .
=• The left unit axiom is equivalent to the isomoprhism´ y a yz  ´ z
´z
´ yz
∼
Jz P az ∼H P
Jz
Jz H a P yz
(a, b) 7→== hom(a, b).
y

b

y

b

b

(192)

The most interesting feature of promonoidal structure in categories isthat they correspond bijectively with monoidal structures on the categoryof functors [C, Sets], heavily generalizing the Day construction of Definition
6.1.
Proposition A.3 : Let P = (P, J, α, ρ, λ) be a promonoidal structure on the
category C; then we can define a P-convolution on the category [C, Sets] (ormore generally, on the category [C, V]), via
ˆ ab[F ∗P G]c =P (a, b; c) × F a × Gb
(193)
JP = J(194)

and this turns out to be a monoidal structure on [C, Sets]. We denote themonoidal structure ([C, Sets], ∗P , JP ) shortly as [C, Sets]P .

Exercise A.4 : Prove the above statement using associativity and unitality
for P.
Exercise A.5 [Day and Cauchy convolutions]: Outline the promonoidal
structure P giving the Day convolution described in Definition 6.1. If C is
any small category, we define P (a, b; c) = C(a, c) × C(b, c) and J to be the
22Obvoiusly, there is nothing special about sets here. The whole discussion performedin the setting of V-profunctors leads to the definition of a V-promonoidal structure.

ENDS

93

terminal functor C → Sets. Outline the convolution product on [C, Sets],
called the Cauchy convolution, obtained from this promonoidal structure.
Definition A.6 : A functor Φ : [A, Sets]P → [B, Sets]Q is said to preserve theconvolution product if the obvious isomorphisms hold in [B, Sets]Q :• Φ(F ∗P G) ∼= Φ(F ) ∗Q Φ(G);• Φ(JP ) = JQ .

Remark A.7 : It is observed in [IK86] that for a monoidal A the categoryof presheaves [Aop , V] endowed with the convolution monoidal structure is
the free monoidal cocompletion of A, having in Mon (monoidal categories,
monoidal functors and monoidal natural transformations) the same universal
b has in Cat.
property that A
Appendix B. Fourier transforms via coends.

Definition B.1 : Let A, C be two promonoidal categories with pormonoidal
structures P and Q respectively; a multiplicative kernel from A to C consistsof a profunctor K : A
C such that there are the two natural isomorphisms´ yz a b yz ´ c c ab
k1 )Ky Kz Px ∼= Kx Pc ;
´c c
∼
k2 )
Kx Jc = Jx .
These isomorphisms say that K “behaves like hom” even if it is not an
endofunctor (in fact, hom is a particular example of a kernel AA,
since the isomorphisms k1 , k2 follow from the ninja Yoneda lemma 2.1).
A multiplicative natural transformation α : K → H is a 2-cell between
profunctors commuting with the structural isomorphisms given in (k1 ), (k2 )in the obvious sense.
Exercise B.2 : Define the category of multiplicative kernels ker(A, C) ⊂Relt(A, C) in such a way that the composition of two kernels is again akernel.
C is a multiplicative kernel
Exercise B.3 : Show that a profunctor K : A
if and only if the cocontinuous functor Lanよ K = K̂ : [A, Sets] → [C, Sets]b under the construction in 5.6 preserves thecorresponding to K̄ : A → C
convolution monoidal structure on both categories [A, Sets]P and [C, Sets]Q .
Describe the isomorphisms k1 , k2 when P is Day convolution.
Exercise B.4 : Show that a functor F : (A, ⊗A , i) → (C, ⊗C , j) between
monoidal categories is strong monoidal, i.e.
• F (a ⊗ b) ∼= F a ⊗ F b;∼
• Fi = jnaturally in a, b if and only if pF = hom(F, 1) is a multiplicative kernel.

94

F. LOREGIAN

Dually, show that for A, C promonoidal, F : C → A preserves convolution
on [A, Sets]P , [C, Sets]Q precisely if pF = hom(1, F ) is a multiplicative kernel.
Definition B.5 : Let K : A
C be a multiplicative kernel between promonoidal
categories; define the K-Fourier transform f 7→ K̂(f ) : C → Sets, obtained as
the image of f : A → Sets under the left Kan extension Lanよ K : [A, Sets] →[C, Sets].
Exercise B.6 : Show the following properties of the K-Fourier transform:• There is the canonical isomorphism
ˆ a
∼
(195)
K̂(f ) =K(a, ) × f (a)• K̂ preserves the convolution monoidal structure (this is the Parseval
identity for the Fourier transform);• K̂ has a right adjoint defined by
ˆ
∼
(196)
Ǩ(g) = [K( , x), g(x)].
x

References[ABLR02] J. Adámek, F. Borceux, S. Lack, and J. Rosickỳ, A classification of accessible
categories, Theory Appl. Categ. (2002), 7–30.
[Ada78] John Frank Adams, Infinite loop spaces, volume 90 of annals of mathematics
studies, Princeton University Press, Princeton, NJ 129 (1978), 247.
Jean Bénabou, Distributors at work, 2000. Lecture notes written by Thomas[Béner]Streicher.
A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations,[BK72]
Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York,1972. MR 0365573
J.L. Borges, Other inquisitions, Texas Pan-American, University of Texas Press,[Bor64]1964.
[Bor94a] F. Borceux, Handbook of categorical algebra. 1, Encyclopedia of Mathematics
and its Applications, vol. 50, Cambridge University Press, Cambridge, 1994,
Basic category theory. MR 1291599 (96g:18001a)[Bor94b], Handbook of categorical algebra. 2, Encyclopedia of Mathematics and its
Applications, vol. 51, Cambridge University Press, Cambridge, 1994, Categories
and structures. MR 1313497 (96g:18001b)[Boz75] Syméon Bozapalides, Les fins cartésiennes, C.R. Acad. Sci. Paris A 281 (1975),
597–600.
[Boz77], Les fins cartésiennes généralisées, Archivum Mathematicum 13 (1977),
no. 2, 75–87.
[Boz80], Some remarks on lax-presheaves, Illinois Journal of Mathematics 24(1980), no. 4, 676–680.
[BZFN10] D. Ben-Zvi, J. Francis, and D. Nadler, Integral transforms and drinfeld centersin derived algebraic geometry, J. Amer. Math. Soc. 23 (2010), no. 4, 909–966.

ENDS

[Car]
[CKS03]95

David Carchedi, Topos theory course.
JRB Cockett, J Koslowski, and RAG Seely, Morphisms and modules for polybicategories, Theory and Applications of Categories 11 (2003), no. 2, 15–74.
[CP90]
J.-M. Cordier and T. Porter, Fibrant diagrams, rectifications and a constructionof Loday, J. Pure Appl. Algebra 67 (1990), no. 2, 111–124. MR 1080880
Jean-Marc Cordier and Timothy Porter, Homotopy coherent category theory,[CP97]
Transactions of the American Mathematical Society 349 (1997), no. 1, 1–54.
[CP08]
J.M. Cordier and T. Porter, Shape theory: Categorical methods of approximation,
Dover books on mathematics, Dover Publications, 2008.
[Day74] B.J. Day, An embedding theorem for closed categories, Category Seminar (Proc.
Sem., Sydney, 1972/1973), Springer, Berlin, 1974, pp. 55–64. Lecture Notes in
Math., Vol. 420. MR 0364396 (51 #650)[Day77] B. J. Day, Note on compact closed categories, J. Austral. Math. Soc. Ser. A 24
(1977), no. 3, 309–311. MR 0470024[Day11] B.J. Day, Monoidal functor categories and graphic Fourier transforms, Theory
Appl. Categ. 25 (2011), No. 5, 118–141. MR 2805747 (2012h:18007)[DF78]
W. Dwyer and E. Dror Farjoun, Homotopy coends, Online preprint: http://www.ma.huji.ac.il/~landau/preprint98/preprint3.ps (1997/8).
Brian J. Day and Gregory M. Kelly, Enriched functor categories, Reports[DK69]of the Midwest Category Seminar, III, Springer, Berlin, 1969, pp. 178–191.
MR 0255633 (41 #293)[DM96] G. De Marco, Analisi 1. primo corso di analisi matematica. teoria ed esercizi,Collana di matematica. Testi e manuali, Zanichelli, 1996.
[Dub70] Eduardo J. Dubuc, Kan extensions in enriched category theory, Lecture Notes
in Mathematics, Vol. 145, Springer-Verlag, Berlin-New York, 1970. MR 0280560(43 #6280)[EK66]
Samuel Eilenberg and G.M. Kelly, A generalization of the functorial calculus, J.
Algebra 3 (1966), 366–375. MR 0190204 (32 #7618)[Elm83] A. D. Elmendorf, Systems of fixed point sets, Trans. Amer. Math. Soc. 277(1983), no. 1, 275–284. MR 690052[EM45]
Samuel Eilenberg and Saunders MacLane, General theory of natural equivalences,
Trans. Amer. Math. Soc. 58 (1945), 231–294. MR 0013131[EP08]
Philip John Ehlers and Timothy Porter, Ordinal subdivision and special pastingin quasicategories, Advances in Mathematics 217 (2008), no. 2, 489–518.
[Gam10] Nicola Gambino, Weighted limits in simplicial homotopy theory, J. Pure Appl.
Algebra 214 (2010), no. 7, 1193–1199. MR 2586997 (2011e:18030)[Gen15] F. Genovese, Quasi-functors as lifts of fourier-mukai functors: the uniquenessproblem, Ph.D. thesis, Università degli studi di Pavia, 2015.
[GHN15] David Gepner, Rune Haugseng, and Thomas Nikolaus, Lax colimits and free
fibrations in ∞-categories, arXiv preprint arXiv:1501.02161 (2015).
P.G. Goerss and J.F. Jardine, Simplicial homotopy theory, Modern Birkhäuser
[GJ09]Classics, Birkhäuser Verlag, Basel, 2009, Reprint of the 1999 edition[MR1711612]. MR 2840650[GL]
F. Genovese and F. Loregian, A general theory of coherent co/ends, In preparation.
[GPS95] R. Gordon, A. J. Power, and Ross Street, Coherence for tricategories, Mem.
Amer. Math. Soc. 117 (1995), no. 558, vi+81. MR 1261589[Gra]
John W. Gray, Closed categories, lax limits and homotopy limits, 127–158.
MR 593251 (82f:18007a)

96[Gra74][Gra80][Gro13][GV72][GZ67][Hof11][Hör14][Hov99][IK86][Isa09][Joy02a][Joy02b][Joy08][JT07][Kel80][Kel82][Kel89][Kel05][KS74][Law73][LB15][Low13]

F. LOREGIAN, Formal category theory: adjointness for 2-categories, Springer-Verlag,
Berlin, 1974, Lecture Notes in Mathematics, Vol. 391. MR MR0371990 (51#8207), Closed categories, lax limits and homotopy limits, J. Pure Appl. Algebra
19 (1980), 127–158. MR 593251 (82f:18007a)M. Groth, Derivators, pointed derivators and stable derivators, Algebraic &Geometric Topology 13 (2013), no. 1, 313–374.
A. Grothendieck and J. Verdier, Théorie des topos ( sga 4, exposés i-vi), Springer
Lecture Notes in Math 269 (1972), 270.
Peter Gabriel and Michel Zisman, Calculus of fractions and homotopy theory,
Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer-Verlag
New York, Inc., New York, 1967. MR 0210125 (35 #1019)
Alexander E Hoffnung, Spans in 2-categories: A monoidal tricategory, arXiv
preprint arXiv:1112.0560 (2011).
Fritz Hörmann, Homotopy limits and colimits in nature—a motivation forderivators, Lecture notes, summer school on derivators, Freiburg, arXiv (2014).
M. Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63,
American Mathematical Society, Providence, RI, 1999. MR 1650134 (99h:55031)
Geun Bin IM and G.M. Kelly, A universal property of the convolution monoidal
structure, Journal of Pure and Applied Algebra 43 (1986), no. 1, 75 – 88.
Samuel B. Isaacson, A note on unenriched homotopy coends, Online preprint:
http://www-home.math.uwo.ca/~sisaacso/PDFs/coends.pdf (2009).
A. Joyal, Quasi-categories and Kan complexes, J. Pure Appl. Algebra 175 (2002),
no. 1-3, 207–222, Special volume celebrating the 70th birthday of Professor Max
Kelly. MR 1935979 (2003h:55026)
André Joyal, Quasi-categories and Kan complexes, Journal of Pure and Applied
Algebra 175 (2002), no. 1, 207–222.
, The theory of quasi-categories and its applications, Citeseer, 2008.
André Joyal and Myles Tierney, Quasi-categories vs Segal spaces, Categories inalgebra, geometry and mathematical physics, Contemp. Math., vol. 431, Amer.
Math. Soc., Providence, RI, 2007, pp. 277–326. MR 2342834 (2008k:55037)
G. Max Kelly, A unified treatment of transfinite constructions for free algebras,
free monoids, colimits, associated sheaves, and so on, Bulletin of the AustralianMathematical Society 22 (1980), no. 01, 1–83.
, Basic concepts of enriched category theory, no. 64, CUP Archive, 1982.
, Elementary observations on 2-categorical limits, Bulletin of the Australian Mathematical Society 39 (1989), 301–317.
, On the operads of J. P. May, Repr. Theory Appl. Categ. (2005), no. 13,
1–13. MR 2177746 (2006f:18005)
Gregory M. Kelly and R. Street, Review of the elements of 2-categories, Category
seminar, Springer, 1974, pp. 75–103.
Bill Lawvere, Metric spaces, generalised logic, and closed categories, Rendicontidel Seminario Matematico e Fisico di Milano, vol. 43, Tipografia Fusi, Pavia,
1973.
David Li-Bland, The stack of higher internal categories and stacks of iterated
spans, arXiv preprint arXiv:1506.08870 (2015).
Zhen Lin Low, Universes for category theory, arXiv preprint arXiv:1304.5227(2013).

ENDS[Lur09]97

J. Lurie, Higher Topos Theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009. MR 2522659 (2010j:18001)[LV12]
Jean-Louis Loday and Bruno Vallette, Algebraic operads, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],
vol. 346, Springer, Heidelberg, 2012. MR 2954392[May72] J.P. May, The geometry of iterated loop spaces, Springer-Verlag, Berlin-New
York, 1972, Lectures Notes in Mathematics, Vol. 271. MR 0420610 (54 #8623b)[May97] J. P. May, Operads, algebras and modules, Operads: Proceedings of RenaissanceConferences (Hartford, CT/Luminy, 1995), Contemp. Math., vol. 202, Amer.
Math. Soc., Providence, RI, 1997, pp. 15–31. MR 1436914[ML98]
S. Mac Lane, Categories for the working mathematician, second ed., Graduate
Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872(2001j:18001)[MLM92] S. Mac Lane and I. Moerdijk, Sheaves in geometry and logic: A first introduction
to topos theory, Springer, 1992.
[Moe95] I. Moerdijk, Classifying spaces and classifying topoi, Lecture Notes in Mathematics, vol. 1616, Springer-Verlag, Berlin, 1995. MR 1440857 (98f:18006)[MSS02] M. Markl, S. Shnider, and J. Stasheff, Operads in algebra, topology and physics,
Mathematical Surveys and Monographs, vol. 96, American Mathematical Society,
Providence, RI, 2002. MR 1898414 (2003f:18011)[Rie14]
E. Riehl, Categorical homotopy theory, New Mathematical Monographs, vol. 24,
Cambridge University Press, Cambridge, 2014. MR 3221774
E. Riehl and D. Verity, The theory and practice of Reedy categories, Theory[RV14]
Appl. Categ. 29 (2014), 256–301. MR 3217884
U. Schreiber, Differential cohomology in a cohesive ∞-topos, ArXiv:1310.7930[Sch13](2013).
[Seg74]
Graeme Segal, Categories and cohomology theories, Topology 13 (1974), 293–312.
MR 0353298[Shu06] Michael Shulman, Homotopy limits and colimits and enriched homotopy theory,
arXiv preprint arXiv:math/0610194 (2006).
[Str76]
Ross Street, Limits indexed by category-valued 2-functors, Journal of Pure andApplied Algebra 8 (1976), no. 2, 149 – 181.
[Str12]
R. Street, Monoidal categories in, and linking, geometry and algebra, Bull. Belg.
Math. Soc. Simon Stevin 19 (2012), no. 5, 769–821. MR 3009017[Tria]
T. Trimble, Towards a doctrine of operads. nLab page.
[Trib]
S. Tringàli, Plots and their applications - part i: Foundations, arXiv preprint
arXiv:1311.3524.
[TV05]
B. Toën and G. Vezzosi, Homotopical algebraic geometry i: Topos theory,
Advances in mathematics 193 (2005), no. 2, 257–372.
[Uni13]
The Univalent Foundations Program, Homotopy type theory: Univalent foundations of mathematics, http://homotopytypetheory.org/book, Institute forAdvanced Study, 2013.
[Vit06]Claudia Centazzo; Enrico M. Vitale, A classification of geometric morphisms
and localizations for presheaf categories and algebraic categories, Journal of
Algebra 303 (2006).
[Wat75] William C. Waterhouse, Basically bounded functors and flat sheaves, Pacific J.
Math. 57 (1975), no. 2, 597–610. MR 0396578

98

F. LOREGIAN[Wei94][Yon60]Charles A. Weibel, An introduction to homological algebra, Cambridge Studies
in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge,
1994. MR 1269324 (95f:18001)
Nobuo Yoneda, On Ext and exact sequences, J. Fac. Sci. Univ. Tokyo Sect. I 8(1960), 507–576 (1960). MR 0225854 (37 #1445)

University of Western Ontario, London, Ontario — Canada, N6A 5B7
E-mail address: floregia@uwo.ca
E-mail address: fosco.loregian@gmail.com†